# Asmptotic of the eigenvalues of Toeplitz matrices with even symbol.

Philippe Rambour

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Philippe Rambour. Asmptotic of the eigenvalues of Toeplitz matrices with even symbol.. 2021. hal-03121284v7

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# Asmptotic of the eigenvalues of Toeplitz matrices with even symbol.

Philippe Rambour

<figure><img src="image_1.png" /></figure>

► To cite this version:

Philippe Rambour. Asmptotic of the eigenvalues of Toeplitz matrices with even symbol.. 2024. hal-03121284v9

HAL Id: hal-03121284

https://hal.science/hal-03121284v9

Preprint submitted on 26 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
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# Asymptotic of the eigenvalues of Toeplitz matrices with even symbol.

Philippe Rambour*

## Abstract

In this paper we consider an interval $[\theta_1, \theta_2] \subset [0, \pi]$ and $f$ a differentiable, periodic and even function sufficiently smooth such that $f(\theta) \in [f(\theta_1, f(\theta_2))] \iff \theta \in [\theta_1, \theta_2]$. Then we obtain a higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix $T_N(f)$ which belong to $[f(\theta_1, f(\theta_2))]$. Also given are three consequences for Toeplitz matrices with trigonometric symbol.

## Mathematical Subject Classification (2020)

Primary 47L80; Secondary 47A08, 47A10, 47A15, 47G30.

**Keywords** Toeplitz matrices, operator eigenvalues.

## 1 Introduction and statement of the main results

For $h \in L^1(\mathbb{T})$ we denote by $T_N(h)$ the Toeplitz matrix of order $N$ with symbol $h$. It is the $(N+1) \times (N+1)$ matrix such that, for $N \ge k, l \ge 0$, $(T_N(h))_{k+1,l+1} = \hat{h}(k-l)$ where $\hat{h}(u)$ is the Fourier coefficient of order $u$ of $h$ ([13, 8]). For $h$ a real valued function the matrix $T_N(h)$ is an Hermitian Toeplitz matrix. We here consider symmetric Toeplitz matrix, which is equivalent to assuming that the symbol $h$ is an even function and we denote by $\lambda_N^{(1)} \le \lambda_N^{(2)} \le \dots \le \lambda_N^{(N+1)}$ the eigenvalues of the $(N+1) \times (N+1)$ matrix $T_N(h)$. This paper addresses the asymptotic behavior of the eigenvalues of $T_N(h)$ as $N$ goes to the infinity. This is a topic which has attracted mathematicians and physicists for a long time. Toeplitz matrices and their relatives emerge in particular in statistic [9] and in statistical physics [3, 4] and the questions about the asymptotic behavior of their spectral characteristics, especially their determinants, eigenvalues, and eigenvectors, are always at the heart of the matter. We refer to the papers [10] for an extensive list of references. According to the first Szegö limit theorem (see [13]) the eigenvalues of $T_N(h)$ are asymptotically distributed as the value of $h$; see [13] for $L^\infty$ symbols, [25] for $L^1$ symbols, and [23],[24] for more general situations. In the Hermitian case extensive works has been done on the search for eigenvalues (or the extreme eigenvalues) of Toeplitz matrices [26, 13, 21, 20, 16, 18, 17] and more recently, for instance,

<footnote>* Université de Paris Saclay, Bâtiment 307; F-91405 Orsay Cedex; tel : 01 69 15 57 28 ; fax 01 69 15 60 19
e-mail : philippe.rambour@universite-paris-saclay.fr</footnote>
# Asymptotic behavior of the eigenvalues of Toeplitz matrices with even symbol

Philippe Rambour*

**Abstract**

In this paper we consider an interval $[\theta_1, \theta_2] \subset ]0, \pi[$ and $f$ a periodic and even function in $C^4([0, 2\pi])$ such that $f(\theta) \in [f(\theta_1), f(\theta_2)] \iff \theta \in [\theta_1, \theta_2]$ and $f \in A^+([\theta_1, \theta_2])$ (resp. $f \in A^-([\theta_1, \theta_2])$, that means $f'(\theta) > 0$ (resp. $f'(\theta) < 0$) for all $\theta \in [\theta_1, \theta_2]$. Then we obtain a higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix $T_N(f)$ as $N \to +\infty$ which belong to $[f(\theta_1), f(\theta_2)]$ (resp. $[f(\theta_2), f(\theta_1)]$).

**Mathematical Subject Classification (2020)**

Primary 47L80 ; Secondary 47A08, 47A10, 47A15, 47G30.

**Keywords** Toeplitz matrices, operator eigenvalues.

# 1 Introduction and statement of the main results

If $\mathbb{T} = R/(2\pi\mathbb{Z})$ and $h \in L^1(\mathbb{T})$, we denote by $T_N(h)$ the Toeplitz matrix of order $N$ with symbol $h$. It is the $(N+1) \times (N+1)$ matrix such that, for $N \ge k, l \ge 0$, $(T_N(h))_{k+1,l+1} = \hat{h}(k-l)$ where $\hat{h}(u)$ is the Fourier coefficient of order $u$ of $h$ ([16, 9]). For a real valued function $h$ the matrix $T_N(h)$ is a Hermitian Toeplitz matrix. We here consider a symmetric Toeplitz matrix, which is equivalent to assuming that the symbol $h$ is an even function and we denote by $\lambda_N^{(1)} \le \lambda_N^{(2)} \le \cdots \le \lambda_N^{(N+1)}$ the eigenvalues of $T_N(h)$. This paper addresses the asymptotic behavior of the eigenvalues of $T_N(h)$ as $N$ goes to infinity. This is a topic which has attracted mathematicians and physicists for a long time. Toeplitz matrices and their relatives emerge in particular in statistic [10, 5] and in statistical physics [3, 4]. It is known from a long time that Toeplitz matrices are useful for providing Green's kernels for studying the solutions of certain differential equations ([27, 23]), and also for discretizing differential operators with finite differences [19]. Finally, these matrices are used in more recent fields, such as Ising models [11] and iso-geometric analysis [15]. The questions about the asymptotic behavior of their spectral characteristics, especially their determinants, eigenvalues, and eigenvectors, are always at the heart of the matter. We refer to the papers [11] for an extensive list of references. According to the first Szegő limit theorem (see [16]) the eigenvalues of $T_N(h)$ are asymptotically distributed

<footnote>*Université de Paris Saclay, Bâtiment 307; F-91405 Orsay Cedex; tel : 01 69 15 57 28 ;
e-mail : philippe.rambour@universite-paris-saclay.fr</footnote>
[5, 7, 6, 2, 12, 11, 19].

The results of theorem 1 are consistent with those of theorem 2.3 of [5]. But the method of proof is different and the first three points of the statement concern functions which are outside the framework of [5]. The corollaries 1, 2 and 3 (which are immediate consequences of Theorem 1) cannot be obtained from the results of [5]. These corollaries give results for Toeplitz matrices with trigonometric symbol, which are useful for statistic applications. On the other hand Theorem 1 indicates that the problem of the eigenvalues of Toeplitz matrices is a local problem, related to the variation of the function which is the symbol of the matrix. Here we denote by $\mathcal{A}^+([0, 2\pi])$ (resp. $\mathcal{A}^-([0, 2\pi])$) the set of even differentiable periodic functions of period $2\pi$, such that $f'(\theta) > 0$ (resp. $f'(\theta) < 0$) for all $\theta$ in $[0, \pi[$.

Here we must consider a function $f \in L^1(\mathbb{T})$. For $[\theta_1, \theta_2] \subset [0, \pi]$ we say that $f \in \mathcal{A}^+([\theta_1, \theta_2])$ (resp. $f \in \mathcal{A}^-([\theta_1, \theta_2])$) if $f$ is a differentiable, periodic and even function such that $f'(\theta) > 0$ (resp. $f'(\theta) < 0$) for all $\theta \in [\theta_1, \theta_2]$ and also $\forall \theta \in [0, \pi] f(\theta) \in [f(\theta_1, f(\theta_2))] \iff \theta \in [\theta_1, \theta_2]$ (resp. $f(\theta) \in [f(\theta_2, f(\theta_1))] \iff \theta \in [\theta_1, \theta_2]$).

For $\nu \ge 0$ we denote by $W^\nu$ the weighted Wiener algebra of all functions $\psi \mathbb{T} \mapsto \mathbb{C}$ whose Fourier coefficients satisfy

$$ \|\psi\|_{\nu} = \sum_{j \in \mathbb{Z}} |\hat{\psi}(j)| (|j+1|)^{\nu} < \infty. $$

Now we define the functions

$$ H(\theta_1, \theta_2) = \frac{f(\theta_1) - f(\theta_2)}{(1 - \cos \theta_1) - (1 - \cos \theta_2)} $$

and

$$ \rho(\theta) = -\frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H(t, \theta)))}{\tan(\frac{t-\theta}{2})} dt + \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H(t, \theta)))}{\tan(\frac{t+\theta}{2})} dt $$

for $\theta_1, \theta_2, \theta$ in $[0, \pi]$. Lastly for an interval $[a, b] \subset ]\theta_1, \theta_2[$ and $N$ an integer we denote by $k_{a,N}$ and $k_{b,N}$ the integers such that : $k_{a,N} = \min\{k|\frac{k\pi}{N+2} \in [a, b]\}$, $k_{b,N} = \max\{k|\frac{k\pi}{N+2} \in [a, b]\}$. Now we can give the following theorem.

**Theorem 1** Let $f$ in $C^4[0, 2\pi]$ such that $f \in \mathcal{A}^+([\theta_1, \theta_2])$ for an interval $[\theta_1, \theta_2] \subset [0, \pi]$. We have the following results.

i) If $[\theta_1, \theta_2] \subset ]0, \pi[$ and $[a, b] \subset ]\theta_1, \theta_2[$ then for $N$ be a sufficiently large integer $T_N(f)$ has $k_{b,N} - k_{a,N} + 1$ eigenvalues in $f([a, b])$. For $k$ an integer in $[k_{a,N}, k_{b,N}]$ the corresponding eigenvalue is

$$ \tilde{\lambda}_N^{(k)} = f\left(\frac{k\pi}{N+2}\right) + \frac{c_1}{(N+2)} + \frac{c_2}{(N+2)^2} + O\left(\frac{1}{(N+2)^3}\right), $$

uniformly in $\tilde{\lambda}$, with

$$ c_1 = f'\left(\frac{k\pi}{N+2}\right) \rho\left(\frac{k\pi}{N+2}\right) $$

and

$$ c_2 = f'\left(\frac{k\pi}{N+2}\right) \rho'\left(\frac{k\pi}{N+2}\right) \rho\left(\frac{k\pi}{N+1}\right) + \frac{1}{2} f''\left(\frac{k\pi}{N+2}\right) \rho\left(\frac{k\pi}{N+2}\right)^2. $$
as the value of $h$; see [16] for $L^\infty$ symbols, [31] for $L^1$ symbols, and [29, 30] for more general situations. In the Hermitian case extensive works has been done on the search for eigenvalues (or the extreme eigenvalues) of Toeplitz matrices [32, 16, 26, 25, 20, 22, 21] and more recently, for instance, [6, 7, 2, 14, 13, 8]. In [6] the authors give an asymptotic expansion of order 2 for the eigenvalues of a Toeplitz matrix with smooth simple-loop symbol, the results of this article are the closest to ours, but the techniques are different. [7] is a good reminder of the various results obtained on the eigenvalues of Toeplitz matrices with polynomial symbol. In [2] M. Barrera, A. Böttcher, S. M. Grudsky and E. A. Maximenko show that the eigenvalues of the matrix $T_N(4(1 - \cos \theta)^2)$ cannot have an asymptotic expansion to order 4. For banded Toeplitz matrices or block symmetric Toeplitz matrices, the reader is referred to [14, 13]. Here we consider real symbol. For complex symbols an alternative to our results is given in [12]. The results of Theorem 1 are consistent with those of Theorem 2.3 of [6]. But the method of proof is different and the statement concern functions which are outside the framework of [6]. On the other hand Theorem 1 indicates that the problem of the eigenvalues of Toeplitz matrices is a local problem, related to the variation of the function which is the symbol of the matrix.

Here we denote by $\mathcal{A}^+([0, 2\pi])$ (resp. $\mathcal{A}^-([0, 2\pi])$) the set of even differentiable periodic functions of period $2\pi$, such that $f'(\theta) > 0$ (resp. $f'(\theta) < 0$) for all $\theta$ in $[0, \pi[$. More generally $[\theta_1, \theta_2] \subset ]0, \pi[$ we say that $f \in \mathcal{A}^+([\theta_1, \theta_2])$ (resp. $f \in \mathcal{A}^-([\theta_1, \theta_2])$) if $f$ is a differentiable, $2\pi$ periodic, and even function such that $f'(\theta) > 0$ (resp. $f'(\theta) < 0$) for all $\theta \in [\theta_1, \theta_2]$ and also $f(\theta) \in [f(\theta_1), f(\theta_2)] \iff \theta \in [\theta_1, \theta_2]$ (resp. $f(\theta) \in [f(\theta_2), f(\theta_1)] \iff \theta \in [\theta_1, \theta_2]$).

For $\nu \ge 0$ we denote by $W^\nu$ the weighted Wiener algebra of all functions $\psi: \mathbb{T} \to \mathbb{C}$ which admits the representation $\psi(t) = \sum_{j \in \mathbb{Z}} \hat{\psi}(j) t^j$ whose Fourier coefficients satisfy

$$ \|\psi\|_{\nu} = \sum_{j \in \mathbb{Z}} |\hat{\psi}(j)| (|j| + 1)^{\nu} < \infty. $$

Now for $f \in \mathcal{A}^+([\theta_1, \theta_2])$ we define the functions

$$ H(\theta', \theta'') = \frac{f(\theta') - f(\theta'')}{(1 - \cos \theta') - (1 - \cos \theta'')} \quad \text{for } (\theta', \theta'') \in [0, \pi]^2 $$

and

$$ \rho(\theta) = \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H(t, \theta)))}{\tan(\frac{t-\theta}{2})} dt - \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H(t, \theta)))}{\tan(\frac{t+\theta}{2})} dt \quad \text{for } \theta \in [\theta_1, \theta_2]. $$

Lastly for an interval $[a, b] \subset [\theta_1, \theta_2]$ and an integer $N$ we denote by $k_{a,N}$ and $k_{b,N}$ the integers such that: $k_{a,N} = \min\{k|\frac{k\pi}{N+2} \in [a, b]\}$, $k_{b,N} = \max\{k|\frac{k\pi}{N+2} \in [a, b]\}$. We have also to define the two functions

$$ c_1(t) = f'(t)\rho(t) $$

and

$$ c_2(t) = f'(t)\rho'(t)\rho(t) + \frac{1}{2}f''(t)\rho(t)^2; $$

Now we can state our main result and an easy consequence.
ii) If $\theta_1 = 0$ and $b < \theta_2 < \pi$ then for a sufficiently large $N$ the matrix $T_N(f)$ has $k_{b,N}$ eigenvalues in $f([0, b])$, with $0 < \lambda_N^{(1)} < \lambda_N^{(2)} < \dots < \lambda_N^{k_{b,N}} \le f(b)$. For $k \in [1, k_{b,N}]$ the corresponding eigenvalue is as in the point i).

iii) If $\theta_2 = \pi$ and $0 < a < \pi$ then for a sufficiently large $N$ the matrix $T_N(f)$ has $N+1-k_{a,N}$ eigenvalues in $f([a, \pi])$ with $f(a) \le \lambda_N^{(k_{a,N})} < \lambda_N^{(k_{a,N}+1)} < \dots < \lambda_N^{N+1} < f(\pi)$. For $k \in [k_{a,N}, N+1]$ the corresponding eigenvalue is as in the point i).

iv) If $\theta_1 = 0, \theta_2 = \pi$ then for a sufficiently large $N$ the matrix $T_N(f)$ has $N+1$ eigenvalues in $f([0, \pi])$ with $f(0) < \lambda_N^{(1)} < \lambda_N^{(2)} < \dots < \lambda_N^{(N+1)} < f(\pi)$. For $k \in [1, N+1]$ the corresponding eigenvalue is as in the point i).

**Remark 1** Similar results to Theorem 1 holds for the case where $f$ in $A^{-}([\theta_1, \theta_2])$ for an interval $[\theta_1, \theta_2] \subset [0, \pi]$.

**Remark 2** In the point i) of Theorem 1 $\tilde{\lambda}_N^{(k)}$ is not necessarily the $k$-th largest eigenvalue of $f$. This is nevertheless true in the cases $0 = \theta_1 < \theta_2 < \pi$, $0 < \theta_1 < \theta_2 = \pi$, or $0 = \theta_1 < \theta_2 = \pi$.

**Remark 3** Revisiting the proof of Theorem 1, one can show that under the assumption $f \in C^3([0, 2\pi])$ we obtain the different point of the theorem with the formula

$$ \tilde{\lambda}_N^{(k)} = f\left(\frac{k\pi}{N+2}\right) + \frac{c_1}{(N+2)} + O\left(\frac{\log N}{(N+2)^2}\right), $$

uniformly in $\tilde{\lambda}$ and $c_1$ as in Theorem 1

The following corollaries are tree interesting consequences of Theorem 1. For these corollaries we recall that a function $c$ is said to be a regular function iff $c$ is a $2\pi$-periodic function, with $c(\theta) > 0$ for all $\theta \in [0, 2\pi]$ and $c \in L^1([0, 2\pi])$.

In the following we denote by $\lambda_{N,\alpha}^{(1)} \le \lambda_{N,\alpha}^{(2)} \le \dots \le \lambda_{N,\alpha}^{(k)} \le \dots \le \lambda_{N,\alpha}^{(N+1)}$ the eigenvalue of the function $h_\alpha : \theta \mapsto |1 - e^{i\theta}|^{2\alpha}c(\theta)$ where $c$ is an even regular function and $\alpha$ a positive real.

**Corollary 1** Consider a real $\alpha \ge 2$ and a function $h_\alpha : \theta \mapsto |1 - e^{i\theta}|^{2\alpha}c(\theta)$ where $c$ is an even regular function such that

* $c \in C^4([0, 2\pi])$

* $c'(\theta) > 0$ for all $\theta \in]0, \pi[$.

Then for a sufficiently large $N$ the matrix $T_N(h_\alpha)$ has $N+1$ eigenvalues such that

$$ \lambda_{N,\alpha}^{(k)} = h_{\alpha}\left(\frac{k\pi}{N+2}\right) + \frac{c_{1,\alpha}}{(N+2)} + \frac{c_{2,\alpha}}{(N+2)^2} + O\left(\frac{1}{(N+2)^3}\right), $$

for $1 \le k \le N+1$ and where

$$ c_{1,\alpha} = h'_{\alpha}\left(\frac{k\pi}{N+2}\right) \rho_{\alpha}\left(\frac{k\pi}{N+2}\right) $$
**Theorem 1** Let $f$ in $C^4[0, 2\pi]$ be such that $f \in A^+([\theta_1, \theta_2])$ for an interval $[\theta_1, \theta_2] \subset ]0, \pi[$. Then for all interval $[a, b] \subset ]\theta_1, \theta_2[$ and for a sufficiently large integer $N$ we have the two following statements.

1. For all eigenvalue $\lambda$ of $T_N(f)$ in $[f(a), f(b)]$ we have a single integer $k$ in $[k_{\theta_1,N}, k_{\theta_1,N}]$ such that $\lambda = \tilde{\lambda}_N^{(k)} + O(\frac{1}{(N+2)^3})$, uniformly in $\lambda$, with

$$ \tilde{\lambda}_N^{(k)} = f\left(\frac{k\pi}{N+2}\right) + \frac{c_1\left(\frac{k\pi}{N+2}\right)}{(N+2)} + \frac{c_2\left(\frac{k\pi}{N+2}\right)}{(N+2)^2}. $$

2. For all $k \in [k_{a,N}, k_{b,N}]$ the matrix $T_N(f)$ has a single eigenvalue $\lambda$ in $[f(\theta_1), f(\theta_2)]$ such that $\lambda = \tilde{\lambda}_N^{(k)} + O(\frac{1}{(N+2)^3})$ uniformly in $k$.

**Remark 1** Similar results to Theorem 1 holds for the case where $f$ in $A^-([\theta_1, \theta_2])$ for an interval $[\theta_1, \theta_2] \subset ]0, \pi[$.

**Remark 2** In Theorem 1 the eigenvalue $\lambda = \tilde{\lambda}_N^{(k)} + O(\frac{1}{(N+2)^2})$ is not necessarily $\lambda_N^{(k)}$.

**Remark 3** If we consider the functions $\psi_\alpha : \theta \mapsto (1 - \cos\theta)^\alpha c(\theta)$ where $\alpha \ge 2$ and $c$ a even positive function such that $c \in C^4[0, 2\pi]$ and $\alpha \sin\theta(1 - \cos\theta)^{\alpha-1}c(\theta) + (1 - \cos\theta)^\alpha c'(\theta) > 0$ for all $\theta \in [0, \pi]$ we can remark that Theorem 1 provides all the eigenvalues of the functions of $\psi_\alpha$ in $[\epsilon, \pi - \epsilon]$ for all $\epsilon > 0$, that is an extension of the main results of [8].

**Remark 4** Revisiting the proof of Theorem 1, one can show that under the assumption $f$ in $C^3([0, 2\pi])$ we obtain an analogous version of this Theorem with the formula $\lambda = \tilde{\lambda}_N^{(k)} + O(\frac{\log N}{(N+2)^2})$, with

$$ \tilde{\lambda}_N^{(k)} = f\left(\frac{k\pi}{N+2}\right) + \frac{c_1\left(\frac{k\pi}{N+2}\right)}{(N+2)} $$

where the function $c_1$ is as in Theorem 1.

Our result can also be compared with that of Trench [28] where it is proved that for this class of symbols the eigenvalues are all distinct.

To conclude we can remark that a tiny modification of the proof of Theorem 1 allows us to obtain the following Theorem which is in fact Theorem 2-3 in [2].

**Theorem 2** Let $f$ in $C^4[0, 2\pi]$ be such that $f \in A^+([0, 2\pi])$, $f''(0) > 0$ and $f''(\pi) < 0$, then for a sufficiently large $N$ we have

$$ \lambda_N^{(k)} = f\left(\frac{k\pi}{N+2}\right) + \frac{c_1\left(\frac{k\pi}{N+2}\right)}{(N+2)} + \frac{c_2\left(\frac{k\pi}{N+2}\right)}{(N+2)^2} + O\left(\frac{1}{(N+2)^3}\right) $$

uniformly in $k = 1, 2, \dots, N$ and with $c_1$ and $c_2$ defined as previously.

Lastly we have to recall the two following definitions

**Definition 1** We denote by $\mathbb{H}^+$ is the set of all functions $\varphi$ in $L^2(\mathbb{T})$ whose Fourier coefficients satisfy $\hat{\varphi}(j) = 0$ for all $j < 0$. $\mathbb{H}^+$ is called the Hardy space on the unit circle.

**Definition 2** In the rest of this paper we denote by $\chi$ the function $\theta \mapsto e^{i\theta}$.
$$c_{2,\alpha} = h'_{\alpha} \left( \frac{k\pi}{N+2} \right) \rho'_{\alpha} \left( \frac{k\pi}{N+2} \right) \rho_{\alpha} \left( \frac{k\pi}{N+1} \right) + \frac{1}{2} h''_{\alpha} \left( \frac{k\pi}{N+2} \right) \rho_{\alpha} \left( \frac{k\pi}{N+2} \right)^2$$

with

$$\rho_{\alpha}(\theta) = -\frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H_{\alpha}(t, \theta)))}{\tan\left(\frac{t-\theta}{2}\right)} dt + \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\ln((H_{\alpha}(t, \theta)))}{\tan\left(\frac{t+\theta}{2}\right)} dt$$

and

$$H_{\alpha}(\theta_1, \theta_2) = \frac{h_{\alpha}(\theta_1) - h_{\alpha}(\theta_2)}{(1 - \cos(\theta_1))(1 - \cos(\theta_2))}$$

for $\theta_1, \theta_2, \theta$ in $[0, \pi]$.

Using the remark 3 we can write the following corollary.

**Corollary 2** Let $\alpha$ be a real such that $\alpha \in [\frac{3}{2}, 2[$ and a function $h_{\alpha}: \theta \mapsto |1 - e^{i\theta}|^{2\alpha}c(\theta)$ where c is an even regular such that

$$\bullet c \in C^3([0, 2\pi])$$

$$\bullet c'(\theta) > 0 \text{ for all } \theta \in ]0, \pi[$$

Then $T_N(h_{\alpha})$ has $N + 1$ eigenvalues in $f([0, \pi])$ with

$$\lambda_{N,\alpha}^{(k)} = h_{\alpha} \left( \frac{k\pi}{N+2} \right) + \frac{c_{1,\alpha}}{(N+2)} + O \left( \frac{\log N}{(N+2)^2} \right),$$

where $N + 1 \ge k \ge 1$ and $c_1$ as in the corollary 1.

Now if we remark that

$$\frac{(1 - \cos \theta_1)c(\theta_1) - (1 - \cos \theta_2)c(\theta_2)}{(1 - \cos \theta_1) - (1 - \cos \theta_2)} = c(\theta_1) + (1 - \cos \theta_2) \frac{c(\theta_1) - c(\theta_2)}{(1 - \cos \theta_1) - (1 - \cos \theta_2)}$$

we obtain the following corollary.

**Corollary 3** Consider a function $h_1: \theta \mapsto |1 - e^{i\theta}|^2 c(\theta)$ where c is an even regular function such that

$$\bullet c \in C^4([0, 2\pi])$$

$$\bullet c'(\theta) > 0 \text{ for all } \theta \in ]0, \pi[$$

Then for all integers $k$ we have

$$\lambda_{N,1}^{(k)} = h_1 \left( \frac{k\pi}{N+2} \right) + \frac{c_1}{(N+2)} + \frac{c_2}{(N+2)^2} + O \left( \frac{1}{(N+2)^3} \right),$$

where $\lambda_{N,1}^{(k)}$ is the $k$-th largest eigenvalue of $T_N(h_1)$ and $c_1, c_2$ as in Corollary 2.

**Remark 4** For $c = 0$ the proof of Theorem 1 provides $c_1 = c_2 = 0$ and we find again the classical result given in [13]

Our result can also be compared with that of Trench [22] where it is proved that for this class of symbols the eigenvalues are all distinct.
## 2 Proof of Theorem 1

### 2.1 Preliminaries

In this proof we have to use the following Theorem which provides an inversion formula for a family of Toeplitz matrices.

**Theorem 3** Let $P_{N+1}$ a trigonometric polynomial with degree $N + 1$ and without zeros on the unit disc $\bar{D}$. Let $\omega = r\bar{\chi}_0$, $0 < r < 1$, $|\chi_0| = 1$ and also $f_r = g_1g_2$, with $g_1 = \chi_0(1 - \omega\chi)(P_{N+1})^{-1}$ and $g_2 = (1 - \omega\bar{\chi})(\bar{P}_{N+1})^{-1}$. Then for all polynomial $P$ in $\mathcal{P}_N = \text{vect}\{1, \chi, \dots, \chi^N\}$ we have

$$T_N(f_r)^{-1}(P) = \frac{1}{g_1}\pi_+\left(\frac{P}{g_2}\right) - \frac{1}{g_1}\pi_+\left(\Phi_N \sum_{s=0}^{+\infty} (H_{\Phi_N}^* H_{\Phi_N})^s \pi_+\left(\tilde{\Phi}_N \pi_+\left(\frac{P}{g_2}\right)\right)\right).$$

with

$$\left\{ \begin{array}{l} \Phi_N = \frac{g_1}{g_2} \chi^{N+1}, \\ \tilde{\Phi}_N = \frac{g_2}{g_1} \chi^{-(N+1)}, \\ H_{\Phi_N}(\Psi) = \pi_-(\Phi_N \Psi), \\ H_{\tilde{\Phi}_N}^*(\tilde{\Psi}) = \pi_+(\tilde{\Phi}_N \Psi) \end{array} \right.$$

for $\Psi \in \mathbb{H}^+$, (resp. $\tilde{\Psi} \in (\mathbb{H}^+)^\perp$) and where $\pi_+$ (resp. $\pi_-$) are the orthogonal projection on $\mathbb{H}^+$ (resp. $(\mathbb{H}^+)^\perp$).

The reader can see [24] for the statement and the proof of Theorem 3. In the appendix of this article we briefly recall how to use it to calculate $((T_N)^{-1}(f_0))_{(1,1)}$ where the symbol $f_0$ is defined by $f_0 = \chi_0(1 - \bar{\chi}_0\chi)(1 - \bar{\chi}_0\bar{\chi}) \frac{1}{|P_{N+1}|^2}$. The equation (28) gives the expression of $((T_N)^{-1}(f_0))_{(1,1)}$. We use this expression to obtain the equation (4) which is a fundamental tool of our proof.

Always to obtain (4) we have to use the fundamental property of the predictor polynomials which is the property (1). Before stating this property, we need of course to recall the definition of the predictor polynomial and its main property.

**Definition 3** The predictor polynomial of degree $M$ of a regular function $h$ is the trigonometric polynomial $K_M$ defined by

$$K_M = \sum_{k=0}^{M} \frac{(T_M(h))_{k+1,1}^{-1}}{\sqrt{(T_M(h))_{1,1}^{-1}}} \chi^k.$$

**Property 1** For all integers $j$, such that $-M \le j \le M$ we have

$$\left( \frac{1}{|K_M|^2} \right) (j) = \hat{h}(j).$$

We have also the useful property

**Property 2** $K_M(re^{i\theta}) \ne 0$ for all $\theta \in \mathbb{R}$ and $0 \le r \le 1$.
## 2 Proof of Theorem 1

In this proof we have used an inversion formula adapted for the Toeplitz matrix with symbol $f = g_1g_2$, $g_1 = \chi_0(1 - r\bar{\chi}_0\chi)(P_{N+1})^{-1}$, $g_2 = (1 - r\bar{\chi}_0\bar{\chi})(\overline{P_{N+1}})^{-1}$, where $0 < r < 1$, and $P_{N+1}$ a polynomial with degree $N + 1$ without zeros on $\bar{D} = \{z/|z| \le 1\}$, $\chi$ is the function $\theta \mapsto e^{i\theta}$, $\chi_0 = e^{i\theta_0}$, with $\theta_0 \in \mathbb{R}$ (see [19] for the statement and the proof of this formula). In the appendix of this article we give this formula and briefly recall how we use it to calculate the inverse of a Toeplitz matrix $(N+1) \times (N+1)$ with symbol $\chi_0(1 - \bar{\chi}_0\chi)(1 - \bar{\chi}_0\bar{\chi}) \frac{1}{|P_{N+1}|^2}$. Now we need to recall the definition of the predictor polynomial and its main property.

**Definition 1** The predictor polynomial of degree $M$ of a regular function $h$ is the trigonometric polynomial $K_M$ defined by

$$K_M = \sum_{k=0}^{M} \frac{(T_M(h))_{k+1,1}^{-1}}{\sqrt{(T_M(h))_{1,1}^{-1}}} \chi^k.$$

**Property 1** For all integers $j$, such that $-M \le j \le M$ we have

$$\left( \frac{1}{|K_M|^2} \right) (j) = \hat{h}(j).$$

For an integer $k$ we denote by $\beta_{k,N}$ the coefficient of $\chi^k$ in the predictor polynomial $P_N$ of $f$. The reader can consult [15] for the predictor polynomials. We will now give the following definition, which will be used in the demonstration.

**Definition 2** We denote by $\mathbb{H}^+$ is the set of all functions $\varphi$ in $L^2(\mathbb{T})$ whose Fourier coefficients satisfy $\hat{\varphi}(j) = 0$ for all $j < 0$.

We can now begin the demonstration of the theorem.

Using the assumptions we can write $f(\theta) = f_1(1 - \cos \theta)$ where $f_1$ is a differentiable function strictly increasing on $[0, 2]$. For all $\lambda$ in $[f(\theta_1), f(\theta_2)]$ we put $\theta_\lambda = f^{-1}(\lambda)$ and $\lambda' = f_1^{-1}(\lambda)$, that means $\theta_\lambda = \arccos(1 - \lambda')$, since $\lambda' \in [0, 2]$ and $\theta_\lambda \in [0, \pi]$.

We have clearly, for $\lambda \in [f(\theta_1), f(\theta_2)]$,

$$f(\theta) - \lambda = f_1(1 - \cos \theta) - \lambda = ((1 - \cos \theta) - (1 - \cos \theta_\lambda)) H_\lambda(\theta)$$

where $H_\lambda : \theta \mapsto H(\theta, \theta_\lambda)$ is a regular function on $[-\pi, \pi]$. We can write

$$((1 - \cos \theta) - (1 - \cos \theta_\lambda)) = ((1 - \cos \theta) - \lambda') = \frac{1}{2} (|1 - \chi|^2 - 2\lambda'). \quad (1)$$

If $\chi_\lambda = e^{i\theta_\lambda}$ and $\chi = e^{i\theta}$ we have $\chi_{\lambda'} = (1 - \lambda') + i\sqrt{1 - (\lambda' - 1)^2}$ and we can write the equation (1) as

$$((1 - \cos \theta) - (1 - \cos \theta_\lambda)) = -\frac{1}{2} \chi_\lambda (1 - \bar{\chi}_\lambda \chi) (1 - \bar{\chi}_\lambda \bar{\chi}). \quad (2)$$

**Remark 5** If we denote by $P_{N+1,\lambda}$ the predictor polynomial of $H_\lambda$ the main property of these polynomials (see the property 1) allows to write that the $(N+1) \times (N+1)$ Toeplitz matrices of the respective symbol $((1 - \cos \theta) - (1 - \cos \theta_\lambda)) H_{\lambda'}$ and $-\frac{1}{2} \chi_{\lambda'}(1 - \bar{\chi}_\lambda \chi)(1 - \bar{\chi}_\lambda \bar{\chi}) \frac{1}{|P_{N+1,\lambda'}|^2}$ are the same.
Finally, if $\mathcal{P}_M$ is the set of trigonometric polynomials of degree less than or equal to $M$ we consider the scalar product defined on $\mathcal{P}_M$ by $\langle P|Q \rangle = \int_0^{2\pi} P(\theta)Q(\theta)h(\theta)d\theta$ and let's denote $\Phi_0, \Phi_1, \dots, \Phi_M$ the orthogonal polynomials for this scalar product. Then the predictor polynomials of degree 0, 1, \dots, $M$ are closely related to these orthogonal polynomials by the relation

$$K_j(z) = z^j \bar{\Phi}_j \left( \frac{1}{z} \right) \quad \forall j, 0 \le j \le M \quad \text{and} \quad \forall z \ne 0.$$

The reader can consult [18] for the predictor polynomials.

We can now begin the demonstration of the theorem.

This demonstration is divided into three parts. In the first part we obtain the equation (10) whose solutions are of the form $f^{-1}(\lambda)$ where the reals $\lambda$ are the eigenvalues of $f$ belonging to $[\theta_1, \theta_2]$. In the second part we obtain an integral expression for the $\rho_N$ functions involved in this equation, which gives us the uniform convergence of the $\rho_N$ in $[\theta_1, \theta_2]$ to a continuous function $\rho$. The $\rho_N$ are therefore uniformly bounded, allowing us to locate the solutions of (10). In the third part, we study the smoothness of the function $\rho$, and use Taylor's theorem in (10) to obtain the asymptotic formula stated in Theorem.

## 2.2 Equation for the eigenvalues

Using the assumptions we can write $f(\theta) = f_1(1 - \cos\theta)$ where $f_1$ is a differentiable function strictly increasing on $[0, 2]$. For all $\lambda$ in $[f(\theta_1), f(\theta_2)]$ we put $\theta_\lambda = f^{-1}(\lambda)$ and $\lambda' = f_1^{-1}(\lambda)$, that means $\theta_\lambda = \arccos(1 - \lambda')$, and $\theta_\lambda \in [0, \pi]$.

**Remark 5** In the next proof we denote by $I_{\theta_1, \theta_2}$ the set $[f(\theta_1), f(\theta_2)]$.

For $\lambda \in I_{\theta_1, \theta_2}$ we have

$$f(\theta) - \lambda = f_1(1 - \cos\theta) - \lambda = ((1 - \cos\theta) - (1 - \cos\theta_\lambda)) H_\lambda(\theta)$$

where $H_\lambda : \theta \mapsto H(\theta, \theta_\lambda)$ is a regular function on $[-\pi, \pi]$. We can write

$$(1 - \cos\theta) - (1 - \cos\theta_\lambda) = (1 - \cos\theta) - \lambda' = \frac{1}{2}(|1 - \chi|^2 - 2\lambda'). \quad (1)$$

If $\chi_\lambda = e^{i\theta_\lambda}$ we have $\chi_\lambda = (1 - \lambda') + i\sqrt{1 - (\lambda' - 1)^2}$ and we can write the equation (1) as

$$(1 - \cos\theta) - (1 - \cos\theta_\lambda) = -\frac{1}{2}\chi_\lambda(1 - \bar{\chi}_\lambda\chi)(1 - \bar{\chi}_\lambda\bar{\chi}). \quad (2)$$

Denote by $P_{N+1,\lambda}$ the predictor polynomial of degree $(N+1)$ $H_\lambda$. The property (1) allows to write the equation

$$T_N((1 - \cos\theta) - (1 - \cos\theta_\lambda)) H_{\lambda'} = T_N\left(-\frac{1}{2}\chi_\lambda(1 - \bar{\chi}_\lambda\chi)(1 - \bar{\chi}_\lambda\bar{\chi})\frac{1}{|P_{N+1,\lambda}|^2}\right). \quad (3)$$

For a fixed integer $N$ we denote by $T_{1,N,\lambda}$ the quantity $((T_N(f) - \lambda I_N)^{-1})_{1,1}$. Since $T_N(f) - \lambda I_N$ is a Toeplitz matrix we have, for $\lambda \notin \text{Spec}(T_N(f))$

$$T_{1,N,\lambda} = \frac{\det(T_{N-1}(f) - \lambda I_{N-1})}{\det(T_N(f) - \lambda I_N)}.$$
For $N$ a fixed integer we denote by $T_{1,N,\lambda}$ the quantity $((T_N(f) - \lambda I_N)^{-1})_{1,1}$. We have

$$T_{1,N,\lambda} = \frac{\det(T_{N-1}(f) - \lambda I_{N-1})}{\det(T_N(f) - \lambda I_N)}$$

and, since the eigenvalues of $(T_{N-1}(f))$ are not in $Spec(T_N(f))$ (see [14, 1]), we have

$$\lambda \in \mathrm{Spec}(T_N(f)) \iff \frac{1}{T_{1,N,\lambda}} = 0.$$

Using the remark (5) and the inversion formula of Toeplitz matrices obtained in [19] (the reader can consult the appendix to have an explicit expression of this formula) we obtain the entry $(T_N^{-1}(f))_{(1,1)}$. Hence with the results of the appendix we can write

$$\frac{1}{T_{1,N,\lambda}} = \frac{1 - \bar{\chi}_{\lambda}^{2(N+1)} \tau_N(\chi_{\lambda})}{(1 - \bar{\chi}_{\lambda}^{2(N+1)} \tau_N(\chi_{\lambda})) B_{2,N,\lambda} - B_{1,N,\lambda}} \quad (3)$$

With

$$\tau_N(\theta_\lambda) = \frac{\bar{P}_{N+1,\lambda}(\bar{\chi}_\lambda) P_{N+1,\lambda}(\bar{\chi}_\lambda)}{\bar{P}_{N+1,\lambda}(\chi_\lambda) P_{N+1,\lambda}(\chi_\lambda)},$$

and

$$B_{1,N,\lambda} = |P_{N+1,\lambda}(0) \frac{P_{N+1,\lambda}(\chi_{\lambda})}{\bar{P}_{N+1,\lambda}(\bar{\chi}_{\lambda})}|^2 (1 - \bar{\chi}_{\lambda'}^2)^{-1}.$$

For $\chi_{\lambda'} \neq -1, 1$ the quantity $B_{1,N,\lambda}$ is defined and the definition of the predictor polynomial provides that $P_{N+1,\lambda}(0) = \frac{\det(T_N(H_\lambda))}{\det(T_{N+1}(H_\lambda))}$. Since $\det(T_N(H_\lambda)) \neq 0$ for all $N \in \mathbb{N}^*$ (see [13]) and $P_{N+1,\lambda}(e^{i\theta}) \neq 0$ for all $\theta \in \mathbb{R}$ (see [15]) we have $B_{1,N,\lambda} \neq 0$ for $\chi_{\lambda'} \neq -1, 1$. The constant $B_{2,N,\lambda}$ will be defined for $\chi_{\lambda} \neq -1, 1$. $\chi_{\lambda} \in \{-1, 1\} \iff \lambda' \in \{0, 2\} \iff \lambda \in \{\min_{\mathbb{T}} f, \max_{\mathbb{T}} f\}$. and with the additional remark that neither $\min f$ nor $\max f$ cannot be eigenvalues of $f$ we can consider that $B_{2,N,\lambda}$ is defined. But in any case this constant does not matter in our demonstration. Indeed if $(1 - \bar{\chi}_{\lambda}^{2(N+2)} \tau_N(\chi_{\lambda})) B_{2,N,\lambda} - B_{1,N,\lambda} = 0$ this equality means that $\det(T_{N-1}(f) - \lambda I_{N-1}) = 0$ and $\lambda$ is an eigenvalue of $T_{N-1}(f)$ so it cannot be an eigenvalue of $T_N(f)$.

Hence we can write

$$\lambda \in (\mathrm{Spec}(T_N(f)) \cap I) \iff \chi_{\lambda}^{2(N+2)} = \tau_N(\theta_{\lambda}), \lambda \in I \quad (4)$$

Since the function $H_\lambda$ is even, the constant $\tau_N(\theta_\lambda)$ can be rewritten as

$$\tau_N(\theta_\lambda) = \left( \frac{P_{N+1,\lambda}(\bar{\chi}_\lambda)}{P_{N+1,\lambda}(\chi_\lambda)} \right)^2.$$

As the function $\theta_\lambda \mapsto \frac{P_{N+1,\lambda}(\bar{\chi}_\lambda)}{P_{N+1,\lambda}(\chi_\lambda)}$ is continuous from $I$ to $\{z || z| = 1\}$ we have a non negative function $\rho_N$ defined and continuous on $I$ such that $\tau_N(\theta_\lambda) = e^{2i\rho_N(\theta_\lambda)}$. Then the equation (4) can be written

$$\lambda \in (\mathrm{Spec}(T_N(f)) \cap I) \iff \chi_{\lambda}^{2(N+2)} = e^{2i\rho_N(\theta_{\lambda})}, \lambda \in I. \quad (5)$$
Using equation (3) and the inversion formula of Toeplitz matrices see in Theorem (3) we obtain the entry $T_{1,N,\lambda}$. Then with the results (see the equation (28) in the appendix) we can write, for $\lambda \notin \text{Spec}(T_N(f))$,

$$ T_{1,N,\lambda} = \frac{(1 - \bar{\chi}_{\lambda}^{2(N+2)} \tau_N(\chi_{\lambda})) B_{2,N,\lambda} - B_{1,N,\lambda}}{1 - \bar{\chi}_{\lambda}^{2(N+1)} \tau_N(\chi_{\lambda})}, \quad (4) $$

and, for $\lambda \notin \text{Spec}(T_{N-1}(f)) \cup \text{Spec}(T_N(f))$ we have

$$ \frac{\det(T_N(f) - \lambda I_{N-1})}{\det(T_{N-1}(f) - \lambda I_N)} = \frac{1 - \bar{\chi}_{\lambda}^{2(N+1)} \tau_N(\chi_{\lambda})}{(1 - \bar{\chi}_{\lambda}^{2(N+2)} \tau_N(\chi_{\lambda})) B_{2,N,\lambda} - B_{1,N,\lambda}}, \quad (5) $$

with

$$ \tau_N(\theta_\lambda) = \frac{\bar{P}_{N+1,\lambda}(\chi_\lambda) P_{N+1,\lambda}(\chi_\lambda)}{\bar{P}_{N+1,\lambda}(\bar{\chi}_\lambda) P_{N+1,\lambda}(\bar{\chi}_\lambda)}, $$

and

$$ B_{1,N,\lambda} = |P_{N+1,\lambda}(0)|^2 \tau_N(\theta_\lambda) \bar{\chi}^{2N+2} (1 - \bar{\chi}_\lambda^2), \quad B_{2,N,\lambda} = \frac{\chi_\lambda}{|P_{N+1,\lambda}(0)|^2}. $$

Property 2 implies that the quantities $B_{1,N,\lambda}$ and $B_{2,N,\lambda}$ are well defined and are different from zero for $\lambda \in I_{\theta_1, \theta_2}$.

Now the applications

$$ \lambda \mapsto \frac{\det(T_N(f) - \lambda I_{N-1})}{\det(T_{N-1}(f) - \lambda I_N)}, \quad \lambda \mapsto \tau_N(\theta_\lambda), \quad \lambda \mapsto B_{1,N,\lambda}, \quad \lambda \mapsto B_{2,N,\lambda}, $$

are continuous on $I_{\theta_1, \theta_2} \setminus \text{Spec}(T_{N-1}(f))$. Hence we can write the equality (5) for all real $\lambda \in I_{\theta_1, \theta_2} \setminus \text{Spec}(T_{N-1}(f))$. Since the eigenvalues of $(T_{N-1}(f))$ are not in $\text{Spec}(T_N(f))$ (see [17, 1]), we can write, according to Property 2:

$$ \lambda \in (\text{Spec}(T_N(f)) \cap I_{\theta_1, \theta_2}) \iff \chi_{\lambda}^{2(N+2)} = \tau_N(\theta_{\lambda}), \lambda \in I_{\theta_1, \theta_2}. \quad (6) $$

Since the function $H_\lambda$ is even, the constant $\tau_N(\theta_\lambda)$ can be rewritten as

$$ \tau_N(\theta_\lambda) = \left( \frac{P_{N+1,\lambda}(\chi_\lambda)}{P_{N+1,\lambda}(\bar{\chi}_\lambda)} \right)^2. $$

As the function $\theta \mapsto \frac{P_{N+1,f(\theta)}(e^{-i\theta})}{P_{N+1,f(\theta)}(e^{i\theta})}$ is continuous from $[\theta_1, \theta_2]$ to $\{z \mid |z| = 1\}$ we have a function $\rho_N$ defined and continuous on $[\theta_1, \theta_2]$ such that $\tau_N(\theta_\lambda) = e^{2i\rho_N(\theta_\lambda)}$. Then equation (6) can be written

$$ \lambda \in (\text{Spec}(T_N(f)) \cap [f(\theta_1), f(\theta_2)]) \iff \theta_\lambda = \frac{\rho_N(\theta_\lambda) + k\pi}{N+2}, k \in [0, 2N+3]. \quad (7) $$

More precisely if $M_N = \max_{\theta \in [\theta_1, \theta_2]} |\rho_N(\theta)|$ and if $\theta_1 < a < b < \theta_2$ we can write, according to the construction of $\chi_\lambda$

$$ \lambda \in (\text{Spec}(T_N(f)) \cap [f(a), f(b)]) \Rightarrow \theta_\lambda = \frac{\rho_N(\theta_\lambda) + k\pi}{N+2}, k \in \left[ \frac{(N+2)a - M_N}{\pi}, \frac{(N+2)b + M_N}{\pi} \right] \quad (8) $$
Now according to the construction of $\chi_\lambda$ we have necessary, if $k_{1,N}$ and $k_{2,N}$ as in the introduction

$$ \lambda \in (\mathrm{Spec}(T_N(f)) \cap I) \iff \left( \lambda' = 1 - \cos \left( \frac{\rho_N(\theta_\lambda) + k\pi}{N+2} \right) \quad \text{for} \quad k \in [k_{1,N}, k_{2,N}] \right), \lambda \in I. \quad (6) $$

Lastly if we consider the equation the equation

$$ \theta_{\lambda} = \frac{\rho_{N}(\theta_{\lambda}) + k\pi}{N + 2} \qquad (7) $$

it is clear that (7) implies (6). Now we have to make a more precise study of the function $\rho_N$. If $s \ge 0$, then every function $f \in A(\mathbb{T}, s)$ without zeros on $\mathbb{T}$ admits a Wiener-Hopf factorization, that is, there exist function $f_+$ and $f_-$ such that $f(e^{i\theta}) = f_+(e^{i\theta})e^{i\gamma\theta}f_-(e^{i\theta})$ with some $\gamma \in \mathbb{Z}$ the index of the factorization. The function $f_+$ (resp. $f_-$) belongs to set $A(\mathbb{T}, s)_+$ (resp. $A(\mathbb{T}, s)_-$) where

$$ A(\mathbb{T}, s)_+ = \left\{ f(e^{i\theta}) = \sum_{j=0}^{+\infty} \hat{f}(j) e^{ij\theta} \mid \|f\|_s < +\infty \right\} $$

and

$$ A(\mathbb{T}, s)_- = \left\{ f(e^{i\theta}) = \sum_{j=0}^{+\infty} \hat{f}(-j) e^{-ij\theta} \mid \|f\|_s < +\infty \right\}. $$

Here we have clearly $\frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} = \frac{1}{P_{N+1,\lambda}(e^{i\theta})} \frac{1}{P_{N+1,\lambda}(e^{-i\theta})}$ and

$$ \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+ = \frac{1}{P_{N+1,\lambda}(e^{i\theta})} \quad \text{and} \quad \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_- = \frac{1}{P_{N+1,\lambda}(e^{-i\theta})}, $$

with index zero. Now it is well known that in the Wiener-Hopf factorization $\left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+$ can be written in the form :

$$ \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+ = \exp \left( \frac{1}{2} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right) + \frac{1}{2\pi i} P.V. \int_{\mathbb{T}} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)}{e^{i\theta} - e^{i\theta}} dv \right) $$

that be rewritten as

$$ \exp \left( \frac{1}{2} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right) - \frac{1}{4\pi i} P.V. \int_0^{2\pi} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)}{\tan \frac{u-\theta}{2}} du + \frac{1}{4\pi} \int_0^{2\pi} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right) du \right). $$

that provides $\frac{P_{N+1,\lambda}(e^{-i\theta})}{P_{N+1,\lambda}(e^{i\theta})} = e^{i\rho_{N,\lambda}(\theta)}$ with

$$ \rho_{N,\lambda}(\theta) = -\frac{1}{4\pi} P.V. \int_0^{2\pi} \frac{\log\left(\frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2}\right)}{\tan\frac{u-\theta}{2}} du + \frac{1}{4\pi} P.V. \int_0^{2\pi} \frac{\log\left(\frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2}\right)}{\tan\frac{u+\theta}{2}} du $$

and finally $\rho_N(\theta_\lambda) = \rho_{N,\lambda}(\theta_\lambda)$. The same methods give, for $G_\lambda = (H_\lambda)_+$

$$ G_{\lambda}(e^{i\theta}) = (H_{\lambda}(\theta))_{+} = \exp \left( \frac{1}{2} \log (H_{\lambda}(v)) + \frac{1}{2\pi i} P.V. \int_{\mathbb{T}} \frac{\log (H_{\lambda}(v))}{e^{iv} - e^{i\theta}} dv \right) $$
and

$$ \theta_{\lambda} = \frac{\rho_N(\theta_{\lambda}) + k\pi}{N + 2}, \quad \frac{k\pi}{N + 2} \in [a, b] \Rightarrow \lambda \in \left( \text{Spec}(T_N(f)) \cap \left[f(a - \frac{M_N}{N+2}), f(b) + \frac{M_N}{N+2}\right] \right). \quad (9) $$

Lastly it is clear that we have now to solve the equation

$$ \theta = \frac{\rho_N(\theta) + k\pi}{(N + 2)}. \quad (10) $$

Now we have to make a more precise study of the function $\rho_N$. If $s \ge 0$, then every function $f \in A(\mathbb{T}, s)$ without zeros on $\mathbb{T}$ admits a Wiener-Hopf factorization, that is, there exist functions $f_+$ and $f_-$ such that $f(e^{i\theta}) = f_+(e^{i\theta})e^{i\gamma\theta}f_-(e^{i\theta})$ with some $\gamma \in \mathbb{Z}$ the index of the factorization. The function $f_+$ (resp. $f_-$) belongs to set $A(\mathbb{T}, s)_+$ (resp. $A(\mathbb{T}, s)_-$) where

$$ A(\mathbb{T}, s)_+ = \{f \in W^s | f(e^{i\theta}) = \sum_{j=0}^{+\infty} \hat{f}(j) e^{ij\theta}\} $$

and

$$ A(\mathbb{T}, s)_- = \{f \in W^s | f(e^{i\theta}) = \sum_{j=0}^{+\infty} \hat{f}(-j) e^{-ij\theta}\}. $$

Here we have clearly $|P_{N+1,\lambda}(e^{i\theta})|^2 = P_{N+1,\lambda}(e^{i\theta})P_{N+1,\lambda}(e^{-i\theta})$ and

$$ \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+ = \frac{1}{P_{N+1,\lambda}(e^{i\theta})} \quad \text{and} \quad \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_- = \frac{1}{P_{N+1,\lambda}(e^{-i\theta})}, $$

with index zero. Now it is well known that in the Wiener-Hopf factorization $\left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+$ can be written in the form

$$ \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right)_+ = \exp \left( \frac{1}{2} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right) + \frac{1}{2\pi i} P.V. \int_{\mathbb{T}} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(z)|^2} \right)}{z - e^{i\theta}} dz \right) $$

that can be rewritten as

$$ \exp \left( \frac{1}{2} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^2} \right) + \frac{1}{4\pi i} P.V. \int_0^{2\pi} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(e^{iu})|^2} \right)}{\tan \frac{u-\theta}{2}} du \right. \\ \left. + \frac{1}{4\pi} \int_0^{2\pi} \log \left( \frac{1}{|P_{N+1,\lambda}(e^{iu})|^2} \right) du \right). $$

That provides $\frac{P_{N+1,\lambda}(e^{i\theta})}{P_{N+1,\lambda}(e^{-i\theta})} = e^{i\rho_{N,\lambda}(\theta)}$ with

$$ \rho_{N,\lambda}(\theta) = \frac{1}{4\pi} P.V. \int_0^{2\pi} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(e^{iu})|^2} \right)}{\tan \frac{u-\theta}{2}} du - \frac{1}{4\pi} P.V. \int_0^{2\pi} \frac{\log \left( \frac{1}{|P_{N+1,\lambda}(e^{iu})|^2} \right)}{\tan \frac{u+\theta}{2}} du $$
and

$$ \rho_{\lambda}(\theta) = -P.V. \frac{1}{4\pi} \int_{0}^{2\pi} \frac{\log(H_{\lambda}(u))}{\tan \frac{u-\theta}{2}} du + P.V. \frac{1}{4\pi} \int_{0}^{2\pi} \frac{\log(H_{\lambda}(u))}{\tan \frac{u+\theta}{2}} du, \\ \rho(\theta_{\lambda}) = \rho_{\lambda}(\theta_{\lambda}). $$

Now we need the following lemmas to relate the two functions $\rho_N$ and $\rho$.

**Lemma 1** For all $\lambda \in [f(\theta_1, f(\theta_2))]$ $\lambda \neq f(0), f(\pi)$ the function $\theta \mapsto H_{\lambda}(\theta)$ is in $C^3([0, 2\pi])$ and for all $j \in \{0, 1, 2, 3\}$ we have a real $K_j$ no depending from $\lambda$ such that $\|H_{\lambda}^{(j)}\|_{\infty} \leq K_j$

**Remark 6** The hypothesis $\lambda \neq f(0), f(\pi)$ corresponds to the fact that the maximum and minimum of the function $f$ cannot be eigenvalues of $T_N(f)$. The values $f(0)$ and $f(\pi)$ are therefore never considered in the proof of Theorem 1.

**Proof of the lemma 1:** If $t = 1 - \cos \theta$ and $t_{\lambda} = 1 - \cos \theta_{\lambda}$ we have to prove that the function $H_{1,\lambda}: t \mapsto \frac{f_1(t)-f_1(t_{\lambda})}{t-t_{\lambda}}$ is in $C^3([0, 2\pi])$ and that for all integer $j$, $0 \le j \le 3$ we have a real $K_{1,j}$ such that, for all $\lambda$ in $[f(\theta_1), f(\theta_2)]$ we have $\|H_{1,\lambda}^{(j)}\|_{\infty} \le K_{1,j}$. Clearly $\|H_{1,\lambda}\|_{\infty} \le \|f^{(1)}\|_{\infty}$. Now we have for $t \neq t_{\lambda}$

$$ H_{1,\lambda}^{(1)}(t) = \frac{f_1^{(1)}(t)(t - t_{\lambda}) - (f_1(t) - f_1(t_{\lambda}))}{(t - t_{\lambda})^2} \\ = \frac{\left(f_1^{(1)}(t_{\lambda}) + (t - t_{\lambda})f_1^{(2)}(a_1)\right)(t - t_{\lambda}) - \left(f_1^{(1)}(t_{\lambda})(t - t_{\lambda}) + \frac{(t - t_{\lambda})^2}{2}f_1^{(2)}(a_2)\right)}{(t - t_{\lambda})^2} $$

with $a_1$ and $a_2$ between $t$ and $t_{\lambda}$. That provides

$$ \bullet H_{1,\lambda}^{(1)}(t_{\lambda}) = \frac{f_1^{(2)}(t_{\lambda})}{2} $$

$$ \bullet \|H_{1,\lambda}^{(1)}\|_{\infty} \le \frac{3}{2} \|f_1^{(2)}\|_{\infty} $$

Now we have, for $t \neq t_{\lambda}$

$$ H_{1,\lambda}^{(2)}(t) = \frac{f_1^{(2)}(t)(t - t_{\lambda})^2 - 2 \left(f_1^{(1)}(t)(t - t_{\lambda}) - (f_1(t) - f_1(t_{\lambda}))\right)}{(t - t_{\lambda})^3} \\ = \frac{\left(f_1^{(2)}(t_{\lambda}) + f_1^{(3)}(a_3)(t - t_{\lambda})\right)(t - t_{\lambda})^2 - 2 \left(d_{1,\lambda}(t) - d_{2,\lambda}(t)\right)}{(t - t_{\lambda})^3}. $$

with

$$ d_{1,\lambda}(t) = f_1^{(1)}(t_{\lambda}(t - t_{\lambda}) + f_1^{(2)}(t_{\lambda})(t - t_{\lambda})^2 + f_1^{(3)}(a_4)\frac{(t - t_{\lambda})^3}{2} \\ d_{2,\lambda}(t) = f_1^{(1)}(t_{\lambda}(t - t_{\lambda}) + f_1^{(2)}(t_{\lambda})\frac{(t - t_{\lambda})^2}{2} + f_1^{(3)}(a_5)\frac{(t - t_{\lambda})^3}{6} $$

and $a_3, a_4, a_5$ between $t$ and $t_{\lambda}$. That provides

$$ \bullet H_{1,\lambda}^{(2)}(t_{\lambda}) = \frac{f_1^{(3)}(t_{\lambda})}{3} $$
and finally $\rho_N(\theta_\lambda) = \rho_{N,\lambda}(\theta_\lambda)$, and $\rho_N(\theta) = \rho_{N,f(\theta)}(\theta)$. The same methods give, for $G_\lambda = (H_\lambda)_+$

$$G_{\lambda}(e^{i\theta}) = (H_{\lambda}(\theta))_{+} = \exp \left( \frac{1}{2} \log (H_{\lambda}(v)) + \frac{1}{2\pi i} P.V. \int_{\mathbb{T}} \frac{\log (H_{\lambda}(v))}{e^{iv} - e^{i\theta}} dv \right)$$

and we obtain

$$\rho_{\lambda}(\theta) = \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\log (H_{\lambda}(u))}{\tan \frac{u-\theta}{2}} du - \frac{1}{4\pi} P.V. \int_{0}^{2\pi} \frac{\log (H_{\lambda}(u))}{\tan \frac{u+\theta}{2}} du,$$

Then we put

$$\rho(\theta_{\lambda}) = \rho_{\lambda}(\theta_{\lambda}), \text{ and } \rho(\theta) = \rho_{f(\theta)}(\theta).$$

## 2.3 Limit of the sequence $(\rho_N)_{(N \in \mathbb{N})}$

Now we need to relate the two functions $\rho_N$ and $\rho$ and for this we have to obtain the following property

**Property 3** When $N$ goes to the infinity $|\rho_N(\theta) - \rho(\theta)| = O(\frac{1}{N^2})$ uniformly in $\theta \in [\theta_1, \theta_2]$.

Lemmas 1 to 4 are devoted to the prove of this property.

**Lemma 1** For all $\lambda \in I_{\theta_1}, \theta_2$ the function $\theta \mapsto H_\lambda(\theta)$ is in $C^3([0, 2\pi])$ and for all $j \in \{0, 1, 2, 3\}$ we have a real $K_j$ not depending on $\lambda$ such that $\|H_\lambda^{(j)}\|_{\infty} \le K_j$.

**Remark 6** Here $\|h\|_{\infty} = \sup_{x \in [0, 2\pi]} |h(x)|$ for all bounded function $h$ on $[0, 2\pi]$.

**Proof of the lemma 1:** If $t = 1 - \cos \theta$ and $t_\lambda = 1 - \cos \theta_\lambda$ we have to prove that the function $H_{1,\lambda}: t \mapsto \frac{f_1(t)-f_1(t_\lambda)}{t-t_\lambda}$ is in $C^3([0, 2\pi])$ and that for all integer $j$, $0 \le j \le 3$ there exists a real $K_{1,j}$ such that, for all $\lambda$ in $I_{\theta_1, \theta_2}$ $\|H_{1,\lambda}^{(j)}\|_{\infty} \le K_{1,j}$. Clearly $\|H_{1,\lambda}\|_{\infty} \le \|f^{(1)}\|_{\infty}$. Now for $t \ne t_\lambda$

$$H_{1,\lambda}^{(1)}(t) = \frac{f_{1}^{(1)}(t)(t-t_{\lambda}) - (f_{1}(t)-f_{1}(t_{\lambda}))}{(t-t_{\lambda})^{2}} \\ = \frac{\left(f_{1}^{(1)}(t_{\lambda}) + (t-t_{\lambda})f_{1}^{(2)}(a_{1})\right)(t-t_{\lambda}) - \left(f_{1}^{(1)}(t_{\lambda})(t-t_{\lambda}) + \frac{(t-t_{\lambda})^{2}}{2}f_{1}^{(2)}(a_{2})\right)}{(t-t_{\lambda})^{2}}$$

with $a_1$ and $a_2$ between $t$ and $t_\lambda$. That provides

$$\bullet H_{1,\lambda}^{(1)}(t_{\lambda}) = \frac{f_{1}^{(2)}(t_{\lambda})}{2},$$

$$\bullet \|H_{1,\lambda}^{(1)}\|_{\infty} \le \frac{3}{2} \|f_{1}^{(2)}\|_{\infty}.$$

Now we have, for $t \ne t_\lambda$

$$H_{1,\lambda}^{(2)}(t) = \frac{f_{1}^{(2)}(t)(t-t_{\lambda})^2 - 2 \left(f_{1}^{(1)}(t)(t-t_{\lambda}) - (f_{1}(t)-f_{1}(t_{\lambda}))\right)}{(t-t_{\lambda})^3} \\ = \frac{\left(f_{1}^{(2)}(t_{\lambda}) + f_{1}^{(3)}(a_{3})(t-t_{\lambda})\right)(t-t_{\lambda})^2 - 2 \left(d_{1,\lambda}(t) - d_{2,\lambda}(t)\right)}{(t-t_{\lambda})^3}$$
$$ \bullet \quad \|H_{1,\lambda}^{(2)}\|_{\infty} \le \frac{7}{3} \|f_1^{(3)}\|_{\infty} $$

Finally we can write, always for $t \ne t_\lambda$

$$ \begin{aligned} H_{1,\lambda}^{(3)}(t) &= \frac{f_1^{(3)}(t)(t-t_\lambda)^3 - 3(f_1^{(2)}(t)(t-t_\lambda)^2 - 2(f_1^{(1)}(t)(t-t_\lambda) - (f_1(t)-f_1(t_\lambda))))}{(t-t_\lambda)^4} \\ &= \frac{(f_1^{(3)}(t_\lambda) + f_1^{(4)}(a_6)(t-t\lambda))(t-t_\lambda)^3 - 3(d_{3,\lambda}(t) - 2d_{4,\lambda}(t))}{(t-t_\lambda)^4} \end{aligned} $$

with

$$ \begin{aligned} d_{3,\lambda}(t) &= f_1^{(2)}(t_\lambda)(t-t_\lambda)^2 + f_1^{(3)}(t_\lambda)(t-t_\lambda)^3 + f_1^{(4)}(a_7)\frac{(t-t_\lambda)^4}{2} \\ d_{4,\lambda}(t) &= f_1^{(1)}(t_\lambda)(t-t_\lambda) + f_1^{(2)}(t_\lambda)(t-t_\lambda)^2 + f_1^{(3)}(t_\lambda)\frac{(t-t_\lambda)^3}{2} + f_1^{(4)}(a_8)\frac{(t-t_\lambda)^4}{6} \end{aligned} $$

and $a_6, a_7, a_8$ between $t$ and $t_\lambda$. This last equalities gives us

$$ \bullet \quad H_{1,\lambda}(t_\lambda) = \frac{1}{4}f_1^{(4)}(t_\lambda) $$

$$ \bullet \quad \|H_{1,\lambda}\|_{\infty} \le \frac{15}{4} \|f_1\|_{\infty}. $$

that ends the proof.

□

**Lemma 2** We have a real H no depending from $k$ and $\lambda$ such that

$$ \left| \frac{1}{G_{\lambda}}(k) \right| \le \frac{H}{k^3}, \quad \text{for } k > 0, \quad \text{and} \quad \left| \frac{\widehat{G_{\lambda}}}{\widehat{G_{\lambda}}}(k) \right| \le \frac{H}{k^3}. \quad \text{for } k \ne 0 $$

**Proof** : We can observe that, for $0 \le j \le 3$ $(\pi_+(\log H_\lambda))^{(j)} = \pi_+((\log H_\lambda)^{(j)})$. Hence, with the lemma 1, we have, for $0 \le j \le 3$,

$$ \| (\pi_+(\log H_\lambda))^{(j)} \|_2 \le \| (\log H_\lambda)^{(j)} \|_2 \le T_j \quad (8) $$

If $m_0$ is the minimum of $H$ on $[0, 2\pi] \times [\theta_1, \theta_2]$ it is clear that for all $j \in \{0, 1, 2, 3\} T_j$ is only depending from the constants $m_0, K_0, K_1, K_2, K_3$, of the lemma 1. Hence $T_j$ is no depending from $\lambda$.

On the other hand since $\log H_\lambda \in C^3([0, 2\pi])$ we have, for all $n \ge 0$

$$ |\widehat{\log H_\lambda}(n)| \le \frac{\|(\log H_\lambda)^{(3)}\|_2}{n^3} \le \frac{T_3}{n^3} $$

and

$$ |(\widehat{\log H_\lambda})^{(1)}(n)| \le \frac{\|(\log H_\lambda)^{(3)}\|_2}{n^2} \le \frac{T_3}{n^2}. $$

Hence

$$ \| \exp(\pi_+(\log H_\lambda)) \|_{\infty} \le \exp \left( T_3 \sum_{n \ge 0} \frac{1}{n^3} \right) = M_1 $$
with

$$ 
\begin{aligned}
d_{1,\lambda}(t) &= f_1^{(1)}(t_\lambda)(t-t_\lambda) + f_1^{(2)}(t_\lambda)(t-t_\lambda)^2 + f_1^{(3)}(a_4)\frac{(t-t_\lambda)^3}{2} \\
d_{2,\lambda}(t) &= f_1^{(1)}(t_\lambda)(t-t_\lambda) + f_1^{(2)}(t_\lambda)\frac{(t-t_\lambda)^2}{2} + f_1^{(3)}(a_5)\frac{(t-t_\lambda)^3}{6},
\end{aligned}
$$

and $a_3, a_4, a_5$ between $t$ and $t_\lambda$. That provides

$$ \bullet H_{1,\lambda}^{(2)}(t_\lambda) = \frac{f_1^{(3)}(t_\lambda)}{3}, $$

$$ \bullet \|H_{1,\lambda}^{(2)}\|_{\infty} \le \frac{7}{3} \|f_1^{(3)}\|_{\infty}. $$

Finally we can write, for $t \ne t_\lambda$

$$ 
\begin{aligned}
H_{1,\lambda}^{(3)}(t) &= \frac{f_1^{(3)}(t)(t-t_\lambda)^3 - 3(f_1^{(2)}(t)(t-t_\lambda)^2 - 2(f_1^{(1)}(t)(t-t_\lambda) - (f_1(t)-f_1(t_\lambda)))\}}{(t-t_\lambda)^4} \\
&= \frac{\left(f_1^{(3)}(t_\lambda) + f_1^{(4)}(a_6)(t-t_\lambda)\right)(t-t_\lambda)^3 - 3(d_{3,\lambda}(t) - 2d_{4,\lambda}(t))}{(t-t_\lambda)^4}.
\end{aligned}
$$

with

$$ 
\begin{aligned}
d_{3,\lambda}(t) &= f_1^{(2)}(t_\lambda)(t-t_\lambda)^2 + f_1^{(3)}(t_\lambda)(t-t_\lambda)^3 + f_1^{(4)}(a_7)\frac{(t-t_\lambda)^4}{2} \\
d_{4,\lambda}(t) &= f_1^{(1)}(t_\lambda)(t-t_\lambda) + f_1^{(2)}(t_\lambda)(t-t_\lambda)^2 + f_1^{(3)}(t_\lambda)\frac{(t-t_\lambda)^3}{2} + f_1^{(4)}(a_8)\frac{(t-t_\lambda)^4}{6}
\end{aligned}
$$

and $a_6, a_7, a_8$ between $t$ and $t_\lambda$. This last equalities give us

$$ \bullet H_{1,\lambda}^{(3)}(t_\lambda) = \frac{1}{4}f_1^{(4)}(t_\lambda), $$

$$ \bullet \|H_{1,\lambda}^{(3)}\|_{\infty} \le \frac{15}{4} \|f_1\|_{\infty}, $$

which end the proof.

□

**Remark 7** If $h$ is a function is $L^2([0, 2\pi])$ we denote by $\|h\|_{q,2}$ the quadratic norm $(\int_0^{2\pi} |h((t))|^2 dt)^{\frac{1}{2}}$.

**Lemma 2** We have a real $S_0$ not depending on $k$ and $\lambda$ such that

$$ |\widehat{G}_{\lambda}(k)| \le \frac{S_0}{k^3}, \quad |\widehat{\overline{G}}_{\lambda}(k)| \le \frac{S_0}{k^3}, \quad \text{for } k > 0, \quad \text{and} \quad |\widehat{\overline{G}}_{\lambda}(k)| \le \frac{S_0}{k^3}, \quad \text{for } k \ne 0. $$

**Proof** : We can observe that, for $0 \le j \le 3$, $(\pi_+(\log H_\lambda))^{(j)} = \pi_+((\log H_\lambda)^{(j)})$. Hence, with Lemma 1, we have, for $0 \le j \le 3$,

$$ \| (\pi_+(\log H_\lambda))^{(j)} \|_{q,2} \le \| (\log H_\lambda)^{(j)} \|_{q,2} \le T_j. \quad (11) $$

If $m_0$ is the minimum of $H$ on $[0, 2\pi] \times [\theta_1, \theta_2]$ it is clear that for all $j \in \{0, 1, 2, 3\}$ $T_j$ is only depending on the constants $m_0, K_0, K_1, K_2, K_3$, of Lemma 1. Hence $T_j$ is no depending from
and

$$ \| (\pi_+(\log H_\lambda))^{(1)} \|_{\infty} \le T_3 \sum_{n \ge 0} \frac{1}{n^2} = M_2. $$

Now if we put $\pi_+(\log H_\lambda) = F_\lambda$ we can write

$$ (\exp -F_{\lambda})^{(3)} = \left( -F_{\lambda}^{(3)} + 3F_{\lambda}^{(1)}F_{\lambda}^{(2)} - (F_{\lambda}^{(1)})^3 \right) \exp F_{\lambda}. $$

Clearly we have the inequalities

$$ \begin{aligned} \| F_{\lambda}^{(3)} \exp F_{\lambda} \|_{2} &\le \| F_{\lambda}^{(3)} \|_{2} \| \exp F_{\lambda} \|_{\infty} \le T_2 M_1 \\ \| F_{\lambda}^{(1)} F_{\lambda}^{(2)} \exp F_{\lambda} \|_{2} &\le \| F_{\lambda}^{(1)} \|_{\infty} \| \exp F_{\lambda} \|_{\infty} \| F_{\lambda}^{(2)} \|_{2} \le M_1 M_2 T_2 \\ \| (F_{\lambda}^{(1)})^3 \exp F_{\lambda} \|_{2} &\le \| (F_{\lambda}^{(1)}) \|_{\infty}^3 \| \exp F_{\lambda} \|_{\infty} \le M_2^3 M_1 \end{aligned} $$

according to (8) this means that $\| (\exp F_\lambda)^{(3)} \|_2$, is bounded by a constant $H_1$ no depending from $\lambda$. This result implies

$$ |\exp -F_{\lambda}(n)| \le \frac{\| (\exp -F_{\lambda})^{(3)} \|_{2}}{n^{3}} \le \frac{H_{1}}{n^{3}} $$

for all $n \ge 0$, and $H$ no depending from $\lambda$ and $n$. On the other hand for $n > 0$ we have

$$ \begin{aligned} \left| \frac{\widehat{G_{\lambda}}(n)}{\widehat{G_{\lambda}}(0)} \right| &= \left| \sum_{h \ge 0} \widehat{G_{\lambda}}(h+n) \frac{1}{\widehat{G_{\lambda}}(h)} \right| \\ &\le H_1^2 \frac{\sum_{h>0} \frac{1}{h^3}}{n^3} + \frac{1}{n^3} \left| \frac{1}{\widehat{G_{\lambda}}(0)} \right|, \end{aligned} $$

and

$$ \begin{aligned} \left| \frac{\widehat{G_{\lambda}}(-n)}{\widehat{G_{\lambda}}(0)} \right| &= \left| \sum_{k \ge 0} \widehat{G_{\lambda}}(k) \frac{1}{\widehat{G_{\lambda}}(k)} \right| \\ &\le H_1^2 \frac{\sum_{k>0} \frac{1}{k^3}}{n^3} + \frac{1}{n^3} \left| \frac{1}{\widehat{G_{\lambda}}(0)} \right|, \end{aligned} $$

with the additional remark that $|\widehat{\frac{1}{G_\lambda}}(0)| \le \frac{1}{m_0}$ with $m_0$ as previous we have the lemma with $H = \max(H_1, H_1^2 \sum_{k>0} \frac{1}{k^3} + \frac{1}{m_0})$, where $m_0 = \min\{H(\theta_1, \theta_2)| (\theta_1, \theta_2) \in [0, 2\pi] \times [f(\theta_1), f(\theta_2)]\}$. $\square$

**Lemma 3** If $\beta_{k,\lambda} = \widehat{\frac{1}{G_\lambda}}(k)$ we have, for a sufficient large $N$

$$ ((T_N((H_\lambda)))_{k,1}^{-1} = \overline{\beta_{0,\lambda}} \beta_{k,\lambda} + R_{k,N,\lambda} $$

with $|R_{k,N,\lambda}| \le \frac{M}{N^2(N+1-k)^2}$ where $M$ is no depending from $\lambda$ and $k$.
$\lambda$.

On the other hand since $\log H_{\lambda} \in C^3([0, 2\pi])$ we have, for all $n \ge 0$

$$|\widehat{\log H_{\lambda}}(n)| \le \frac{\|(\log H_{\lambda})^{(3)}\|_{q,2}}{n^3} \le \frac{T_3}{n^3}$$

and

$$|\widehat{(\log H_{\lambda})^{(1)}(n)}| \le \frac{\|(\log H_{\lambda})^{(3)}\|_{q,2}}{n^2} \le \frac{T_3}{n^2}.$$

Hence

$$\| \exp(\pi_+(\log H_{\lambda})) \|_{\infty} \le \exp \left( T_3 \sum_{n \ge 0} \frac{1}{n^3} \right) = M_1 \quad (12)$$

and

$$\| (\pi_+(\log H_{\lambda}))^{(1)} \|_{\infty} \le T_3 \sum_{n \ge 0} \frac{1}{n^2} = M_2.$$

Now if we put $\pi_+(\log H_{\lambda}) = F_{\lambda}$ we can write

$$(\exp -F_{\lambda})^{(3)} = \left( -F_{\lambda}^{(3)} + 3F_{\lambda}^{(1)}F_{\lambda}^{(2)} - (F_{\lambda}^{(1)})^3 \right) \exp -F_{\lambda},$$

and

$$(\exp F_{\lambda})^{(3)} = \left( F_{\lambda}^{(3)} + 3F_{\lambda}^{(1)}F_{\lambda}^{(2)} + (F_{\lambda}^{(1)})^3 \right) \exp F_{\lambda}.$$

According to (11) we have the inequalities,

$$\|F_{\lambda}^{(3)} \exp F_{\lambda}\|_{q,2} \le \|F_{\lambda}^{(3)}\|_{q,2} \|\exp F_{\lambda}\|_{\infty} \le T_2 M_1,$$

$$\|F_{\lambda}^{(1)}F_{\lambda}^{(2)} \exp F_{\lambda}\|_{q,2} \le \|F_{\lambda}^{(1)}\|_{\infty} \|\exp F_{\lambda}\|_{\infty} \|F_{\lambda}^{(2)}\|_{q,2} \le M_1 M_2 T_2,$$

$$\|(F_{\lambda}^{(1)})^3 \exp F_{\lambda}\|_{q,2} \le \|(F_{\lambda}^{(1)})^3\|_{\infty} \|\exp F_{\lambda}\|_{\infty} \le M_2^3 M_1.$$

This means that $\|(\exp F_{\lambda})^{(3)}\|_2$, is bounded by a constant $S_1$ not depending on $\lambda$ and $n$. This result implies

$$|\widehat{\exp -F_{\lambda}}(n)| \le \frac{\|(\exp -F_{\lambda})^{(3)}\|_{q,2}}{n^3} \le \frac{S_1}{n^3}$$

and

$$|\widehat{\exp F_{\lambda}}(n)| \le \frac{\|(\exp F_{\lambda})^{(3)}\|_{q,2}}{n^3} \le \frac{S_1}{n^3}$$

for all $n \ge 0$, that is the first part of the lemma.

On the other hand for $n > 0$ we have

$$\begin{aligned} \left| \frac{G_{\lambda}}{\widehat{G}_{\lambda}}(n) \right| &= \left| \sum_{h \ge 0} \widehat{G}_{\lambda}(h+n) \frac{1}{\widehat{G}_{\lambda}}(-h) \right| \\ &\le S_1^2 \frac{\sum_{h>0} \frac{1}{h^3}}{n^3} + \frac{1}{n^3} \left| \frac{1}{\widehat{G}_{\lambda}}(0) \right|, \end{aligned}$$
**Proof** : Using the inversion formula given in the appendix of this paper we obtain, for $H_\lambda = G_\lambda \bar{G}_\lambda$, $G_\lambda \in \mathbb{H}^+(\mathbb{T})$.

$$ (T_N(H_\lambda))_{l+1,k+1}^{-1} = \left\langle \pi_+ \left( \frac{\chi^l}{\bar{G}_\lambda} \right) \middle| \frac{\chi^k}{\bar{G}_\lambda} \right\rangle - \left\langle \sum_{s=0}^{+\infty} \left( H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}} \right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^l}{\bar{G}_\lambda} \right) \middle| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_\lambda} \right) \right\rangle. $$

with

$$ \Phi_{N,\lambda} = \frac{G_\lambda}{\bar{G}_\lambda} \chi^{N+1}, \quad \text{and} \quad \tilde{\Phi}_{N,\lambda} = \frac{\bar{\lambda}}{G_\lambda} \chi^{-(N+1)} $$

and (see the appendix for $H^+, \pi_+, \pi_-$)

$$ H_{\Phi_{N,\lambda}}(\Psi) = \pi_-(\Phi_{N,\lambda}\Psi) \quad \text{for} \quad \Psi \in \mathbb{H}^+ $$

$$ H_{\Phi_{N,\lambda}}^*(\Psi) = \pi_+(\tilde{\Phi}_{N,\lambda}\Psi) \quad \text{for} \quad \Psi \in (\mathbb{H}^+)^\perp. $$

For $l=0$ this formula becomes

$$ (T_N(H_\lambda))_{k+1,1}^{-1} = \left\langle \pi_+ \left( \frac{1}{\bar{G}_\lambda} \right) \middle| \frac{\chi^k}{\bar{G}_\lambda} \right\rangle - \left\langle \sum_{s=0}^{+\infty} \left( H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}} \right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_\lambda} \right) \middle| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_\lambda} \right) \right\rangle. $$

In the next of the proof we use the following notations :

$$ \frac{1}{G_\lambda} = \sum_{u \ge 0} \beta_{u,\lambda} \chi^u \quad \frac{G_\lambda}{\bar{G}_\lambda} = \sum_{u \in \mathbb{Z}} \gamma_{u,\lambda} \chi^u. $$

With lemma 2 we have a positive constant $H$ such that

$$ |\beta_{u,\lambda}| \le \frac{H}{u^3} \quad \forall u \in \mathbb{N}^* \quad \text{and} \quad |\gamma_{u,\lambda}| \le \frac{H}{u^3} \quad \forall u \in \mathbb{Z}^*. $$

With these notations we obtain

$$ \left\langle \pi_+ \left( \frac{1}{\bar{G}_\lambda} \right) \middle| \frac{\chi^k}{\bar{G}_\lambda} \right\rangle = \bar{\beta}_{0,\lambda} \beta_{k,\lambda} $$

$$ \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_\lambda} \right) = \pi_+ \left( \bar{\Phi}_{N,\lambda} \bar{\beta}_{0,\lambda} \right) = \bar{\beta}_{0,\lambda} \sum_{v \ge N+1} \bar{\gamma}_{-v,\lambda} \chi^{v-N-1}, $$

$$ \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_\lambda} \right) = \sum_{w=0}^{k} \bar{\beta}_{w,\lambda} \left( \sum_{v \ge N+1-k+w} \bar{\gamma}_{-v,\lambda} \chi^{v-N-1+k-w} \right). $$

Hence, with lemma 2 we obtain

$$ \| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_\lambda} \right) \|_2 \le H (N+1)^{-2} $$

and

$$ \| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_\lambda} \right) \|_2 \le H((N+1-k)^{-2}). $$
and

$$
\begin{aligned}
\left| \frac{\widehat{G_{\lambda}}}{\widehat{G_{\lambda}}}(-n) \right| &= \left| \sum_{k \ge 0} \widehat{G_{\lambda}}(k) \frac{1}{\widehat{G_{\lambda}}} - (k+n) \right| \\
&\le S_1^2 \frac{\sum_{k>0} \frac{1}{k^3}}{n^3} + \frac{1}{n^3} \left| \frac{1}{\widehat{G_{\lambda}}}(0) \right|,
\end{aligned}
$$

and with (12) we can write $|\frac{1}{\widehat{G}_{\lambda}}(0)| \le \frac{1}{2\pi} \|\exp(\pi_+(\log H_{\lambda}))\|_{\infty} \le \frac{M_1}{2\pi}$ that provides the third inequality of the lemma. $\square$

**Lemma 3** If $\beta_{k+1,\lambda} = \frac{1}{\widehat{G}_{\lambda}}(k)$ we have, for a sufficient large $N$

$$((T_N(H_\lambda))_{k,1}^{-1} = \bar{\beta}_{0,\lambda}\beta_{k,\lambda} + R_{k,N,\lambda}$$

with $|R_{k,N,\lambda}| \le \frac{M}{N^2(N+1-k)^2}$ where $M$ is not depending on $\lambda$ and $k$.

**Proof** : Using the inversion formula given in the appendix of this paper we obtain, for $H_{\lambda} = G_{\lambda}\bar{G}_{\lambda}$, $G_{\lambda} \in \mathbb{H}^{+}$,

$$(T_N(H_\lambda))_{k+1,l+1}^{-1} = \left\langle \pi_+ \left(\frac{\chi^l}{\bar{G}_\lambda}\right) \left| \frac{\chi^k}{\bar{G}_\lambda} \right\rangle - \left\langle \sum_{s=0}^{+\infty} \left(H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}}\right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left(\frac{\chi^l}{\bar{G}_\lambda}\right) \left| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left(\frac{\chi^k}{\bar{G}_\lambda}\right) \right\rangle, $$

with

$$\Phi_{N,\lambda} = \frac{G_{\lambda}}{\bar{G}_{\lambda}} \chi^{N+1}, \quad \text{and} \quad \tilde{\Phi}_{N,\lambda} = \frac{\bar{G}_{\lambda}}{G_{\lambda}} \chi^{-(N+1)},$$

$$H_{\Phi_{N,\lambda}}(\Psi) = \pi_{-}(\Phi_{N,\lambda}\Psi) \quad \text{for} \quad \Psi \in \mathbb{H}^{+},$$

$$H_{\Phi_{N,\lambda}}^{*}(\Psi) = \pi_{+}(\tilde{\Phi}_{N,\lambda}\Psi) \quad \text{for} \quad \Psi \in (\mathbb{H}^{+})^{\perp}.$$

For $l=0$ this formula becomes

$$(T_N(H_\lambda))_{k+1,1}^{-1} = \left\langle \pi_+ \left(\frac{1}{\bar{G}_\lambda}\right) \left| \frac{\chi^k}{\bar{G}_\lambda} \right\rangle - \left\langle \sum_{s=0}^{+\infty} \left(H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}}\right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left(\frac{1}{\bar{G}_\lambda}\right) \left| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left(\frac{\chi^k}{\bar{G}_\lambda}\right) \right\rangle. $$

In the next of this proof we use the following notation :

$$\frac{G_{\lambda}}{\bar{G}_{\lambda}} = \sum_{u \in \mathbb{Z}} \gamma_{u, \lambda} \chi^u.$$

From Lemma 2 we have a positive constant $S_0$ such that

$$|\beta_{u,\lambda}| \le \frac{S_0}{u^3} \quad \forall u \in \mathbb{N}^* \quad \text{and} \quad |\gamma_{u,\lambda}| \le \frac{S_0}{u^3} \quad \forall u \in \mathbb{Z}^*.$$

We obtain

$$\left\langle \pi_{+} \left(\frac{1}{\bar{G}_{\lambda}}\right) \left| \frac{\chi^{k}}{\bar{G}_{\lambda}} \right\rangle = \bar{\beta}_{0,\lambda} \beta_{k,\lambda}, \right.$$
On the other hand for $\psi = \sum_{w \ge 0} \alpha_w \chi^w$ a function in $\mathbb{H}^+$ we have, with the continuity of the projection $\pi_-$,

$$H_{\Phi_{N,\lambda}}(\psi) = \sum_{w \ge 0} \alpha_w \left( \sum_{v>N+1+w} \gamma_{-v,\lambda} \chi^{v+w+N+1} \right)$$

that provides

$$\begin{aligned} \|H_{\Phi_{N,\lambda}}(\psi)\|_2 &\le \sum_{w \ge 0} |\alpha_w| \left( \sum_{v>N+1+w} |\gamma_{-v,\lambda}| \right) \\ &\le \|\psi\|_2 \left( \sum_{w \ge 0} \left( \sum_{v>N+1+w} |\gamma_{-v,\lambda}| \right)^2 \right)^{1/2} \\ &\le H \|\psi\|_2 (N+1)^{-3/2} \end{aligned}$$

that means $\|H_{\Phi_{N,\lambda}}\| \le H(N+1)^{-3/2}$. Clearly we have also $\|H_{\Phi_{N,\lambda}}^*\| \le H(N+1)^{-3/2}$ and we can write

$$\left\| \sum_{s=0}^{+\infty} \left( H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}} \right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_{\lambda}} \right) \right\|_2 \le \frac{H}{(1-H^2(N+1)^{-3})^2} (N+1)^{-2}.$$

And finally we can write

$$(T_N(H_\lambda))_{1,k+1}^{-1} = \bar{\beta}_{0,\lambda}\beta_{k,\lambda} + O\left((N+1)^{-2}(N+1-k)^{-2}\right)$$

with $O((N+1)^{-2}(N+1-k)^{-2}) = 2H^2(N+1)^{-2}(N+1-k)^{-2}$ uniformly in $\lambda$ that is the expected result with $M = 2H^2$. $\square$

**Remark 7** As the coefficient $\beta_{0,\lambda}$ is real the form of $\tau_N(\chi_\lambda)$ allows to assume that $\beta_{0,\lambda} = 1$ is the rest of our demonstration.

**Lemma 4** For a sufficiently large $N$ we have two constants, no depending from $\lambda$ such that

$$\left\| \frac{1}{P_{N+1,\lambda}} \right\|_0 \le A_1 \quad \text{and} \quad \left\| G_{\lambda} \right\|_0 \le A_2$$

**Proof** : As previously we have $\|P_{N+1,\lambda}^{-1}\|_0 \le \sum_{u=1}^{+\infty} \frac{1}{u^2} \|(P_{N+1,\lambda}^{-1})^{(2)}\|_2$.

We write

$$(P_{N+1,\lambda}^{-1})^{(2)} = \frac{D_{N+1,\lambda}}{P_{N+1,\lambda}^3}.$$

The lemmas 2 and 3 provide that $|D_{N+1,\lambda}|$ is bounded by a constant no depending from $\lambda$ and $\theta$. Since $G_\lambda((\theta) \ne 0$ we have a real $m_1 > 0$ such that for all $\theta \in [0, 2\pi]$ and $\lambda \in [f(\theta_1), f(\theta_2)]$ we can write $|G_\lambda(\theta)| > m_1$. That implies, with the lemma 3, that we have a real $m'_1 > 0$, no depending from $\theta$ and $\lambda$ and such that $\frac{1}{|P_{N+1,\lambda}(e^{i\theta})|^3} < m'_1$, that provides the constant $A_1$.

Lastly we can obtain, as in the lemma 2, $|\hat{G}_\lambda(k)| \le \frac{M_0}{k^3}$ with $M_0$ a constant no depending from $\lambda$, that provides the existence of the real $A_2$. $\square$
$$ \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_{\lambda}} \right) = \pi_+ (\bar{\Phi}_{N,\lambda} \bar{\beta}_{0,\lambda}) = \bar{\beta}_{0,\lambda} \sum_{v \ge N+1} \bar{\gamma}_{-v,\lambda} \chi^{v-N-1}, $$

$$ \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_{\lambda}} \right) = \sum_{w=0}^{k} \bar{\beta}_{w,\lambda} \left( \sum_{v \ge N+1-k+w} \bar{\gamma}_{-v,\lambda} \chi^{v-N-1+k-w} \right). $$

Hence we obtain

$$ \left\| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_{\lambda}} \right) \right\|_{q,2} \le S_1 (N+1)^{-2}, $$

and

$$ \left\| \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{\chi^k}{\bar{G}_{\lambda}} \right) \right\|_{q,2} \le S_1 ((N+1-k)^{-2}), $$

where $S_1$ no depending on $\lambda$ and $N$ On the other hand for $\psi = \sum_{w \ge 0} \alpha_w \chi^w$ a function in $\mathbb{H}^+$ we have, with the continuity of the projection $\pi_-$,

$$ H_{\Phi_{N,\lambda}}(\psi) = \sum_{w \ge 0} \alpha_w \left( \sum_{v > N+1+w} \gamma_{-v,\lambda} \chi^{-v+w+N+1} \right) $$

that provides

$$ \begin{aligned} \| H_{\Phi_{N,\lambda}}(\psi) \|_{q,2} & \le \sum_{w \ge 0} |\alpha_w| \left( \sum_{v > N+1+w} |\gamma_{-v,\lambda}| \right) \\ & \le \|\psi\|_2 \left( \sum_{w \ge 0} \left( \sum_{v > N+1+w} |\gamma_{-v,\lambda}| \right)^2 \right)^{1/2} \\ & \le S_0 \|\psi\|_2 (N+1)^{-3/2} \end{aligned} $$

Clearly we have also $\|H_{\Phi_{N,\lambda}}^*(\psi)\|_{q,2} \le S_0(N+1)^{-3/2}\|(\psi)\|_{q,2}$ and we can write

$$ \left\| \sum_{s=0}^{+\infty} \left( H_{\Phi_{N,\lambda}}^* H_{\Phi_{N,\lambda}} \right)^s \pi_+ \bar{\Phi}_{N,\lambda} \pi_+ \left( \frac{1}{\bar{G}_{\lambda}} \right) \right\|_{q,2} \le \frac{S_1}{(1 - S_0^2(N+1)^{-3})^2} (N+1)^{-2}. $$

And finally we can write

$$ (T_N(H_\lambda))_{1,k+1}^{-1} = \bar{\beta}_{0,\lambda} \beta_{k,\lambda} + O((N+1)^{-2}(N+1-k)^{-2}) $$

with, for a sufficiently large $N$, $O((N+1)^{-2}(N+1-k)^{-2}) = 2S_1^2(N+1)^{-2}(N+1-k)^{-2}$ uniformly in $\lambda$ that is the expected result. $\square$

**Remark 8** As the coefficient $\beta_{0,\lambda}$ is real the form of $\tau_N(\chi_\lambda)$ allows to assume that $\beta_{0,\lambda} = 1$ is the rest of our demonstration.

**Lemma 4** We have $\left\| \ln\left(\frac{1}{|P_{N+1,\lambda}|^2}\right) - \ln(H_\lambda) \right\|_0 = O(\frac{1}{N^2})$ uniformly in $\lambda$.
**Lemma 5** For $s_0 = 0$ we have $\| \ln(\frac{1}{|P_{N+1,\lambda}|^2}) - \ln(H_\lambda) \|_{s_0} = O(\frac{1}{N^2})$ uniformly in $\lambda$.

**Proof** : Using the lemma 3, we obtain

$$ \|P_{N+1,\lambda} - \frac{1}{G_{\lambda}}\|_{s_0} \le M(N+1)^{-2} \sum_{k=0}^{N} \frac{1}{(N+1-k)^2} + \sum_{k=N+1}^{+\infty} |\beta_{k,\lambda}| $$

Hence

$$ \|P_{N+1,\lambda} - \frac{1}{G_{\lambda}}\|_{s_0} \le \frac{M+H}{(N+1)^2} \qquad (9) $$

where $M$ and $\beta_{k,\lambda}$ as in the lemma 3 and $H$ is the real no depending from $N$ and from $\lambda$ which has been introduced in the lemma 2. always with $M$ and $H$ no depending from $\lambda$ and the norm $\|P_{N+1,\lambda} - \frac{1}{G_{\lambda}}\|_{s_0}$ is bounded by $O(\frac{1}{N^2})$. Now since $\|\Psi\Phi\|_{s_0} \le \|\Psi\|_{s_0}\|\Phi\|_{s_0}$ we have

$$ \left\| \frac{1}{P_{N+1,\lambda}} - G_{\lambda} \right\|_{s_0} \le \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{s_0} \left\| \frac{G_{\lambda}}{P_{N+1,\lambda}} \right\|_{s_0} \le A \frac{M+H}{(N+1)^2} \qquad (10) $$

with $A = A_1A_2, A_1, A_2$ as in the lemma 4.

Then, according to (10)

$$ \left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_{s_0} \le \left\| \frac{1}{|P_{N+1,\lambda}|^2} - \frac{1}{P_{N+1,\lambda}} \bar{G}_{\lambda} \right\|_{s_0} + \left\| \frac{1}{P_{N+1,\lambda}} \bar{G}_{\lambda} - H_{\lambda} \right\|_{s_0} \le (A_1+A_2) \left( \frac{M+H}{(N+1)^2} \right) $$

Since $\left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_{s_0} = O(\frac{1}{N^2})$, uniformly in $\lambda$. Now observe that

$$ \left\| \ln \left( \frac{1}{|P_{N+1,\lambda}|^2} \right) - \ln(H_{\lambda}) \right\|_{s_0} = \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_{s_0} $$

that is also

$$ \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_{s_0} \le \sum_{n \ge 1} \frac{1}{n} \left( \left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_{s_0} \right)^n \left( \left\| \frac{1}{H_{\lambda}} \right\|_{s_0} \right)^n . $$

Now we have

$$ \left\| \frac{1}{H_{\lambda}} \right\|_{s_0} \le \frac{1}{\|H_{\lambda}\|_{s_0}} \le \frac{1}{|\widehat{H}_{\lambda}(0)|} \le \frac{1}{m_0} $$

with $m_0$ as in the lemma 2. That gives us

$$ \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_{s_0} \le \sum_{n \ge 1} \frac{1}{n} \left( \left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_{s_0} \right)^n \frac{1}{m_0^n} . $$

Since $m_0$ no depending from $\lambda$ we can conclude

$$ \left\| \ln \left( \frac{1}{|P_{N+1,\lambda}|^2} \right) - \ln(H_{\lambda}) \right\|_{s_0} = \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_{s_0} = O \left( \frac{1}{N^2} \right) $$
Proof : Using Lemma 3, we obtain

$$ \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \leq M(N+1)^{-2} \sum_{k=0}^{N} \frac{1}{(N+1-k)^2} + \sum_{k=N+1}^{+\infty} |\beta_{k,\lambda}|. $$

Hence

$$ \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \leq \frac{M + S_{0}}{(N+1)^{2}}, \qquad (13) $$

where M and $\beta_{k,\lambda}$ as in Lemma 3 and $S_0$ is the real not depending on N and from $\lambda$ which has been introduced in Lemma 2. Always with M and $S_0$ no depending from $\lambda$ and the norm $\|P_{N+1,\lambda} - \frac{1}{G_{\lambda}}\|_0$ is bounded by $O(\frac{1}{N^2})$. Now since $\|\Psi\Phi\|_0 \le \|\Psi\|_0\|\Phi\|_0$ we have

$$ \left\| \frac{1}{P_{N+1,\lambda}} - G_{\lambda} \right\|_{0} \le \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \left\| \frac{G_{\lambda}}{P_{N+1,\lambda}} \right\|_{s_0} \le \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \left\| \frac{1}{P_{N+1,\lambda}} \right\|_{0} \left\| G_{\lambda} \right\|_{0}. \quad (14) $$

Then, according to (14) we have

$$ \left\| \frac{1}{P_{N+1,\lambda}} \right\|_{0} - \left\| G_{\lambda} \right\|_{0} \le \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \left\| \frac{1}{P_{N+1,\lambda}} \right\|_{0} \left\| G_{\lambda} \right\|_{0} \quad (15) $$

That provides

$$ \left\| \frac{1}{P_{N+1,\lambda}} \right\|_{0} \left( 1 - \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \left\| G_{\lambda} \right\|_{0} \right) \le \|G_{\lambda}\|_{0} \quad (16) $$

According to Lemma 2 we have a real $A_1$ such that for all $\lambda$ in $]f(\theta_1, f(\theta_2[ we have $\|G_{\lambda}\|_0 \le A_1$. Hence with (13) we obtain that for N sufficiently large we have

$$ 1 - \left\| P_{N+1,\lambda} - \frac{1}{G_{\lambda}} \right\|_{0} \left\| G_{\lambda} \right\|_{0} \ge \frac{1}{2}. \quad (17) $$

and

$$ \left\| \frac{1}{P_{N+1,\lambda}} \right\|_{0} \le 2A_{1}. \quad (18) $$

Merging (13) and (18) we obtain

$$ \left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_{0} \le \left\| \frac{1}{|P_{N+1,\lambda}|^2} - \frac{1}{P_{N+1,\lambda}} \bar{G}_{\lambda} \right\|_{0} + \left\| \frac{1}{P_{N+1,\lambda}} \bar{G}_{\lambda} - H_{\lambda} \right\|_{0} \le 4A_1^3 \left( \frac{M + S_0}{(N+1)^2} \right) \quad (19) $$

that means $\left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_0 = O(\frac{1}{N^2})$, uniformly in $\lambda$. Now observe that

$$ \left\| \ln \left( \frac{1}{|P_{N+1,\lambda}|^2} \right) - \ln(H_{\lambda}) \right\|_0 = \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_0, $$

that is also

$$ \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_0 \le \sum_{n \ge 1} \frac{1}{n} \left( \left\| \frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda} \right\|_0 \right)^n \left( \left\| \frac{1}{H_{\lambda}} \right\|_0 \right)^n. $$
uniformly in $\lambda$.

Then the fact that the Cauchy singular operator is bounded on the Wiener classes $A(\mathbb{T}, s)$, $s \ge 0$, with $\|\rho_N - \rho\|_{s_0} = O(\frac{1}{N^2})$ that implies that $|\rho_N(\theta_\lambda) - \rho(\theta_\lambda)| = O(\frac{1}{N^2})$ uniformly in $\lambda$. Hence we have two reals $m, M$ with $M \ge m$ such that $m \le \rho_N(\theta) \le M$ for all $\theta \in [\theta_1, \theta_2]$ and all $N \in \mathbb{N}$.

Then for an integer $k \in \{k_{a,N}, \dots, k'_{b,N}\}$ we have, for a sufficiently large $N$, and for all $\theta \in [\theta_1, \theta_2]$,

$$ \theta_1 < \frac{k\pi + m}{N + 2} < \frac{k\pi + \rho_N(\theta)}{N + 2} < \frac{k\pi + M}{N + 2} < \theta_2 $$

and we have two reals $\theta_0$ and $\theta'_0$ such that $\theta_0 < \frac{k\pi+m}{N+2} < \frac{k\pi+\rho_N(\theta)}{N+2}$ and $\frac{k\pi+\rho_N(\theta)}{N+2} < \frac{k\pi+M}{N+2} < \theta'_0$.

Hence the function $\Psi_k : \theta_\lambda \mapsto \frac{\rho_N(\theta_\lambda)+k\pi}{N+2} - \theta_\lambda$ changes sign on $[\theta_1, \theta_2]$ then, by continuity of the function $\rho_N$ on this interval, the equation 7 has at less one solution on this interval.

If $\theta_1 = 0$ we can remark that we have necessary, for all $k \in \{1, \dots, k'_{2,N}\}$ and for all $\theta \in [0, \theta_2]$ the inequality

$$ 0 \le \left( 1 - \cos \left( \frac{\rho_N(\theta) + k\pi}{N + 2} \right) \right) $$

and always :

$$ \left( 1 - \cos \left( \frac{\rho_N(\theta) + k\pi}{N + 2} \right) \right) \le 1 - \cos \left( \frac{k\pi + M}{N + 2} \right) \le 1 - \cos \theta'_0. $$

Hence we can conclude that the equation (6) has at less one solution on $[0, \theta_2]$.

If $\theta_2 = \pi$ we can remark that we have necessary, for all $k \in \{k'_{1,N}, \dots, \pi\}$ and for all $\theta \in [\theta_1, \pi]$ the inequality

$$ \left( 1 - \cos \left( \frac{\rho_N(\theta) + k\pi}{N + 2} \right) \right) \le 2 $$

and always :

$$ 1 - \cos \theta_0 < 1 - \cos \left( \frac{k\pi + m}{N + 2} \right) < 1 - \cos \left( \frac{k\pi + \rho_N(\theta)}{N + 2} \right). $$

Hence we can conclude that the equation (6) has at less one solution on $[\theta_0, \pi]$.

Finally we have obtain the eigenvalues in the interval $[f(a), f(b)]$. Now we have to obtain the asymptotic behavior of these eigenvalues and to prove that we have $k_{b,N} - k_{a,N} + 1$ eigenvalues in this interval Now we need the two following lemmas :

**Lemma 6** The function $\rho$ is in $C^2([\theta_1, \theta_2])$.

**Proof** : We prove the result for the function

$$ I : \theta \mapsto P.V. \int_0^{2\pi} \frac{\ln(H(t, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt, $$

the proof is quite the same for the function

$$ \theta \mapsto P.V. \int_0^{2\pi} \frac{\ln(H(t, \theta))}{\tan\left(\frac{t+\theta}{2}\right)} dt. $$
Now we have, according to Lemma 1,

$$ \left\| \frac{1}{H_{\lambda}} \right\|_{0} \le \left( \sum_{n \ge 0} \frac{1}{n^2} \right) \frac{1}{2\pi} \left\| \left( \frac{1}{H_{\lambda}} \right)^{(2)} \right\|_{q, 2} \le \frac{K}{m_0^3} $$

with $m_0$ as in the proof of Lemma 2 and $K$ no depending on $\lambda$ and $N$. That gives us, according to (19)

$$ \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_0 \le \sum_{n \ge 1} \frac{1}{n} \left( \left\| 4A_1^3 \left( \frac{M+S_0}{(N+1)^2} \right) \right\|_0 \right)^n \left( \frac{K}{m_0^3} \right)^m . $$

Since $m_0$ and $K$ are not depending on $\lambda$ we can conclude

$$ \left\| \ln \left( \frac{1}{|P_{N+1,\lambda}|^2} \right) - \ln(H_{\lambda}) \right\|_0 = \left\| \ln \left( 1 + \frac{\frac{1}{|P_{N+1,\lambda}|^2} - H_{\lambda}}{H_{\lambda}} \right) \right\|_0 = O \left( \frac{1}{N^2} \right) $$

uniformly in $\lambda$.

□

Since the Cauchy singular operator is bounded on the Wiener classes $A(\mathbb{T}, s)$, $s \ge 0$, we have $\|\rho_N - \rho\|_0 = O(\frac{1}{N^2})$ and $|\rho_N(\lambda) - \rho(\lambda)| = O(\frac{1}{N^2})$ uniformly in $\lambda$. That ends the proof of Property 3.

## 2.4 Derivation and solutions of the equation for the eigenvalues

To do this we need the two following lemmas.

**Lemma 5** The function $\rho$ is in $C^2([\theta_1, \theta_2])$.

*Proof* : We prove the result for the function

$$ I : \theta \mapsto P.V. \int_0^{2\pi} \frac{\ln(H(t, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt, $$

the proof is quite the same for the function

$$ \theta \mapsto P.V. \int_0^{2\pi} \frac{\ln(H(t, \theta))}{\tan\left(\frac{t+\theta}{2}\right)} dt. $$

First we write $I(\theta) = I_{1,\theta} + I_{2,\theta}$ with

$$ I_{1,\theta} = P.V. \int_0^{2\pi} \frac{\ln(H(\theta, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt, $$

$$ I_{2,\theta} = \int_{0}^{2\pi} \frac{\log(H(t, \theta)) - \log(H(\theta, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt. $$

A simple calculus provides us $I_{1,\theta} = 0$. On the other hand we can observe that the function $\Psi$ defined by

$$ \Psi(t, \theta) = \frac{\log(H(t, \theta)) - \log(H(\theta, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} $$
First we write $I(\theta) = I_{1,\theta} + I_{2,\theta}$ with

$$I_{1,\theta} = P.V. \int_{0}^{2\pi} \frac{\ln(H(\theta, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt,$$

$$I_{2,\theta} = \int_{0}^{2\pi} \frac{\log(H(t, \theta)) - \log(H(\theta, \theta))}{\tan\left(\frac{t-\theta}{2}\right)} dt.$$

A simple calculus provides us $I_{1,\theta} = 0$. On the other hand we can observe that the function $\Psi : (t, \theta) \mapsto \frac{\log(H(t,\theta)) - \log(H(\theta,\theta))}{\tan\left(\frac{t-\theta}{2}\right)}$ can be written as $\frac{\log(H(t,\theta)) - \log(H(\theta,\theta))}{\frac{t-\theta}{2} \tan\left(\frac{t-\theta}{2}\right)}$. Thanks to the symmetry of the function $H$ we can say that the function $\theta \mapsto H(t,\theta)$ is in $C^3([\theta_1, \theta_2])$ for all $t$ in $[0, 2\pi]$. Hence if $\Psi_1$ is the function defined by $\Psi_1 : \theta \mapsto \frac{\log(H(t,\theta)) - \log(H(\theta,\theta))}{t-\theta}$ the function $\frac{\partial \Psi_1}{\partial \theta}(t, \theta)$ is defined for all $\theta \neq t$ and is equal to

$$\frac{(t - \theta) ((\log H)'_{\theta}(t, \theta) - (\log H)'_{t}(\theta, \theta) - (\log H)'_{\theta}(\theta, \theta)) + ((\log H)(t, \theta) - (\log H)(\theta, \theta))}{(t - \theta)^2}$$

where we have denoted by $(\log H)'_t$ the quantity $\frac{\partial(\log H)}{\partial t}$ and by $(\log H)'_{\theta}$ the quantity $\frac{\partial(\log H)}{\partial \theta}$. We see that for $t = \theta$ the function $\frac{\partial \Psi_1}{\partial \theta}$ is equal to $\frac{\partial^2 \log H}{\partial \theta^2}(\theta, \theta)$. Since the functions $\log H$, $\frac{\partial(\log H)}{\partial t}$, $\frac{\partial(\log H)}{\partial \theta}$, and $\frac{\partial^2 \log H}{\partial \theta^2}$ are continuous on $[0, 2\pi] \times [\theta_1, \theta_2]$ we can conclude that the function $(t, \theta) \mapsto \frac{\partial \Psi}{\partial \theta}$ is defined and continuous on $[0, 2\pi] \times [\theta_1, \theta_2]$, that conclude this demonstration for the existence of $\rho^{(1)}$. For $\rho^{(2)}$ the function $\frac{\partial \Psi_1}{\partial \theta}(t, \theta)$ which is defined for all $t \neq \theta$ and is

$$\frac{(t - \theta)^2 \Psi_2(t, \theta) + 2(-\theta) \Psi_3(t, \theta) + 2 \Psi_3(t, \theta)}{(t - \theta)^3}$$

where, with same notation as for $\rho'$,

$$\Psi_1(t, \theta) = (\log H)''_{\theta_2}(t, \theta) - (\log H)''_{t_2}(\theta, \theta) - (\log H)''_{\theta_2}(\theta, \theta) - 2(\log H)''_{t,\theta}(\theta, \theta)$$

$$\Psi_2(t, \theta) = (\log H)'_{\theta}(t, \theta) - (\log H)'_{t}(\theta, \theta) - (\log H)'_{\theta}(\theta, \theta)$$

$$\Psi_3(t, \theta) = (\log H)(t, \theta) - (\log H)(\theta, \theta)$$

and for $t = \theta$ we see that

$$\frac{\partial^2 \Psi_1}{\partial \theta^2} = (\log H)_{\theta^2, t}^{(3)}(\theta, \theta) + (\log H)_{t^3}^{(3)}(\theta, \theta) + (\log H)_{t^2 \theta}^{(3)}(\theta, \theta)(\theta, \theta).$$

Then the same arguments as previously allow us to conclude the proof. □

Now we can easily obtain the formula announced in the statement of the Theorem 1. For $\lambda$ an eigenvalue in $]f(\theta_1), f(\theta_2)]$ we have, following the equation (7) $\lambda = f\left(\frac{k\pi + \rho_N(\theta_\lambda)}{N+2}\right)$ that is also $\lambda = f\left(\frac{k\pi + \rho(\theta_\lambda)}{N+2} + \frac{R_{N,\lambda}}{(N+2)}\right)$ with $\frac{R_{N,\lambda}}{(N+2)} = O\left(\frac{1}{(N+2)^3}\right)$ uniformly in $\lambda$ according to the lemma 5. Putting $d = \frac{\pi k}{N+2}$ we have by Taylor's theorem,

$$\lambda = f(d) + f'(d) \left( \frac{\rho(\theta_\lambda) + R_{N,\lambda}}{N+2} \right) + \frac{1}{2} f''(d) \left( \frac{\rho(\theta_\lambda) + R_{N,\lambda}}{N+2} \right)^2 \\ + \frac{1}{6} f^{(3)} \left( d + h_1 \frac{\rho(\theta_\lambda) + R_{N,\lambda}}{N+2} \right) \left( \frac{\rho(\theta_\lambda) + R_{N,\lambda}}{N+2} \right)^3,$$
can be write

$$ \Psi(t, \theta) = \frac{\log(H(t, \theta)) - \log(H(\theta, \theta))}{\frac{t-\theta}{2}} \frac{\frac{t-\theta}{2}}{\tan\left(\frac{t-\theta}{2}\right)}. $$

Thanks to the symmetry of the function $H : \theta \mapsto H(\theta, \theta')$ we can say that the function $\theta \mapsto H(t, \theta)$ is in $C^3([\theta_1, \theta_2])$ for all $t$ in $[0, 2\pi]$. Hence if $\Psi_1$ is the function defined by $\Psi_1 : \theta \mapsto \frac{\log(H(t, \theta)) - \log(H(\theta, \theta))}{t-\theta}$ the function $\frac{\partial \Psi_1}{\partial \theta}(t, \theta)$ is defined for all $\theta \neq t$ and is equal to

$$ \frac{(t - \theta) ((\log H)'_{\theta}(t, \theta) - (\log H)'_{t}(\theta, \theta) - (\log H)'_{\theta}(\theta, \theta)) + ((\log H)(t, \theta) - (\log H)(\theta, \theta))}{(t - \theta)^2}, $$

where we have denoted by $(\log H)'_t$ the quantity $\frac{\partial(\log H)}{\partial t}$ and by $(\log H)'_{\theta}$ the quantity $\frac{\partial(\log H)}{\partial \theta}$. We see that for $t = \theta$ the function $\frac{\partial \Psi_1}{\partial \theta}$ is equal to $\frac{\partial^2 \log H}{\partial t^2}(\theta, \theta)$. Since the functions $\log H$, $\frac{\partial(\log H)}{\partial t}$, $\frac{\partial(\log H)}{\partial \theta}$, and $\frac{\partial^2 \log H}{\partial \theta^2}$ are continuous on $[0, 2\pi] \times [\theta_1, \theta_2]$ we obtain that the function $(t, \theta) \mapsto \frac{\partial \Psi_1}{\partial \theta}$ is defined and continuous on $[0, 2\pi] \times [\theta_1, \theta_2]$, that completes this demonstration for the existence of $\rho^{(1)}$. For $\rho^{(2)}$ the function $\frac{\partial^2 \Psi_1}{\partial \theta^2}(t, \theta)$ is defined for all $t \neq \theta$ and is equal to

$$ \frac{(t - \theta)^2 \Psi_1(t, \theta) + 2(t - \theta) \Psi_3(t, \theta) + 2 \Psi_4(t, \theta)}{(t - \theta)^3} $$

where,

$$ \Psi_2(t, \theta) = (\log H)''_{\theta^2}(t, \theta) - (\log H)''_{t^2}(\theta, \theta) - (\log H)''_{\theta^2}(\theta, \theta) - (\log H)''_{t, \theta}(\theta, \theta), $$

$$ \Psi_3(t, \theta) = (\log H)'_{\theta}(t, \theta) - (\log H)'_{t}(\theta, \theta) - (\log H)'_{\theta}(\theta, \theta), $$

$$ \Psi_4(t, \theta) = (\log H)(t, \theta) - (\log H)(\theta, \theta), $$

and for $t = \theta$ we see that

$$ \frac{\partial^2 \Psi_1}{\partial \theta^2} = (\log H)_{\theta^2, t}^{(3)}(\theta, \theta) + (\log H)_{t^3}^{(3)}(\theta, \theta) + (\log H)_{t^2 \theta}^{(3)}(\theta, \theta). $$

Then the same arguments as previously allow us to conclude. □

To begin stating Theorem 1 we have to remark that with Property 3 we have a real $M > 0$ such that $-M \leq \rho_N(\theta) \leq M$ for all integer $N$ and all $\theta \in [\theta_1, \theta_2]$.

Now if $\lambda \in [f(a), f(b)]$ is an eigenvalue of $T_N(f)$ we know that there is a real $\theta_\lambda \in [a, b]$ such that $\theta_\lambda$ is a solution of (10) that implies $a - \frac{M}{N+2} \leq \frac{k\pi}{N+2} \leq b + \frac{M}{N+2}$, and we can conclude $k \in [k_{\theta_1,N}, k_{\theta_2,N}]$ for $N$ sufficiently large.

Reciprocally if $N$ is sufficiently large we have for all $k \in [k_{a,N}, k_{b,N}]$ two reals $\theta'_k$ and $\theta''_k$ in $[\theta_1, \theta_2]$ such that $\theta'_k < \frac{k\pi - M}{N+2}$ and $\frac{k\pi + M}{N+2} < \theta''_k$ that provides a solution to the equation $\theta = \frac{\rho_N(\theta) + k\pi}{N+2}$.

Now we can obtain the formula announced in the statement of Theorem 1. For $\lambda$ an eigenvalue in $]f(\theta_1), f(\theta_2)[$ we have, following the equation (10), $\lambda = f(\frac{k\pi + \rho_N(\theta_\lambda)}{N+2})$ where $\theta_\lambda$ is a solution of the equation (10). According to Property 3 we can enunciate

**Property 4** $\lambda = f(\frac{k\pi + \rho(\theta_\lambda)}{N+2} + \frac{R_{N,\lambda}}{(N+2)})$ with $R_{N,\lambda} = O(\frac{1}{(N+2)^2})$ uniformly in $\lambda$.
with $0 < h_1 < 1$. That provides

$$ \lambda = f(d) + f'(d) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right) + \frac{1}{2} f''(d) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right)^2 + \frac{1}{6} f^{(3)} \left( d + h_1 \frac{\rho(\theta_{\lambda})}{N+2} \right) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right)^3 + O \left( \frac{1}{(N+2)^3} \right), \quad (11) $$

where the quantity $O(\frac{1}{(N+2)^3})$ is bounded uniformly in $\lambda$. On the other hand, with the equation (7) $\theta_{\lambda} = \frac{k\pi + \rho(\theta_{\lambda})}{N+2} + \frac{R_{N,\lambda}}{N^2}$ and we can write, always by Taylor's theorem,

$$ \rho(\theta_{\lambda}) = \rho(d) + \rho'(d) \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} + \frac{1}{2} \rho'' \left( (d + h_2 \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2}) \left( \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right)^2 \right), \quad (12) $$

with $0 < h_2 < 1$, that implies

$$ \rho(\theta_{\lambda}) = \rho(d) + \rho'(d) \frac{\rho(d)}{N+2} + O\left(\frac{1}{(N+2)^2}\right). \quad (13) $$

with

$$ O\left(\frac{1}{(N+2)^2}\right) = \frac{(\rho'(d))^2 \rho(\theta_{\lambda})}{(N+2)^2} + \frac{1}{2(N+2)} \rho''\left((d + h_2 \frac{\rho(\theta_{\lambda})}{N+2}) \left(\frac{\rho(\theta_{\lambda})}{N+2}\right)^2\right) \\ + \frac{1}{2} \rho''\left((d + h_2 \frac{\rho(\theta_{\lambda})}{N+2}) \left(\frac{\rho(\theta_{\lambda})}{N+2}\right)^2\right) + \tilde{R}_{N,\lambda}, $$

where the quantity $\tilde{R}_{N,\lambda}$ can be bounded with $R_{N,\lambda}$, and the maximum of the functions $\rho, \rho'$ and $\rho''$. Combining the equation (11) and (13) we obtain

$$ \lambda = f(d) + \frac{f'(d)\rho(d)}{N+2} + \frac{f'(d)\rho'(d)\rho(d)}{(N+2)^2} + \frac{1}{2}\frac{f''(d)\rho^2(d)}{(N+2)^2} + R_{N,d}, \quad (14) $$

with

$$ \begin{aligned} R_{N,d} &= \frac{f'(d) \left( \rho(\theta_{\lambda}) - \rho(d) - \rho'(d) \frac{\rho(d)}{N+2} \right)}{(N+2)^3} + \frac{f''(d)}{(N+2)^2} \left( \rho(\theta_{\lambda}) - \rho(d) \right)^2 \\ &\quad + \frac{1}{6} f^{(3)} \left( d + h_1 \frac{\rho(\theta_{\lambda})}{N+2} \right) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right)^3 + O\left( \frac{1}{(N+2)^3} \right) \\ &= O\left( \frac{1}{(N+2)^3} \right) \end{aligned} $$

uniformly in $\lambda$. To achieve the proof of the theorem we have to be sure that the eigenvalues found are distinct. To do this we need the following two lemmas.

**Lemma 7** For $k, k+1$ in $]f(\theta_1), f(\theta_2)]$ we have $\tilde{\lambda}_N^{(k+1)} - \tilde{\lambda}_N^{(k)} > 0$ and $|\tilde{\lambda}_N^{(k)} - \tilde{\lambda}_N^{(k+1)}| = O(\frac{1}{N})$

*Proof:* This lemma follows directly from (14). $\square$

**Lemma 8** For a fixed $k$ the equation (7) has one and only one solution in $[\theta_1, \theta_2]$.

*Proof:* Assume $\lambda_N$ and $\lambda'_N$ two solutions of (7) for a same integer $k$. By (14) we have $|\lambda - \lambda'| = o(\frac{1}{(N+2)^2})$. By [1] we know that we have an eigenvalue, $\lambda_{N+1}$ with the bound $\lambda_N < \lambda_{N+1} < \lambda'_N$ that implies $|\lambda_N - \lambda_{N+1}| = o(\frac{1}{(N+2)^2})$. By (14) we have $|\lambda_N - \lambda_{N+1}| \ge O(\frac{1}{N})$, that is a contradiction with the previous estimation. $\square$
Putting $d = \frac{\pi k}{N+2}$ we have by Taylor's theorem,

$$ 
\begin{aligned}
\lambda = f(d) + f'(d) \left( \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right) + \frac{1}{2} f''(d) \left( \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right)^2 \\
+ \frac{1}{6} f^{(3)} \left( d + h_1 \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right) \left( \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right)^3,
\end{aligned}
$$

with $0 < h_1 < 1$. That provides

$$ \lambda = f(d) + f'(d) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right) + \frac{1}{2} f''(d) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right)^2 + \frac{1}{6} f^{(3)} \left( d + h_1 \frac{\rho(\theta_{\lambda})}{N+2} \right) \left( \frac{\rho(\theta_{\lambda})}{N+2} \right)^3 + O \left( \frac{1}{(N+2)^3} \right), \quad (20) $$

where the quantity $O\left(\frac{1}{(N+2)^3}\right)$ is bounded uniformly in $\lambda$. On the other hand, with the equation (10) $\theta_{\lambda} = \frac{k\pi + \rho(\theta_{\lambda})}{N+2} + \frac{R_{N,\lambda}}{(N+2)^2}$ and we can write, always by Taylor's theorem,

$$ \rho(\theta_{\lambda}) = \rho(d) + \rho'(d) \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} + \frac{1}{2} \rho'' \left( (d + h_2 \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2}) \left( \frac{\rho(\theta_{\lambda}) + R_{N,\lambda}}{N+2} \right)^2 \right), \quad (21) $$

with $0 < h_2 < 1$, that implies

$$ \rho(\theta_{\lambda}) = \rho(d) + \rho'(d) \frac{\rho(d)}{N+2} + O\left(\frac{1}{(N+2)^2}\right). \quad (22) $$

with the rest is bounded by $\frac{|S|}{(N+2)^2}$ where $S$ is a constant no depending from $\lambda$. Merging the equation (20) and (22) we obtain

$$ \lambda = f(d) + \frac{f'(d)\rho(d)}{N+2} + \frac{f'(d)\rho'(d)\rho(d)}{(N+2)^2} + \frac{1}{2}\frac{f''(d)\rho^2(d)}{(N+2)^2} + R_{N,d}, \quad (23) $$

with $R_{N,d} = O\left(\frac{1}{(N+2)^3}\right)$ uniformly in $\lambda$. To achieve the proof we have to be sure that the eigenvalues found are distincts as announced. To do this we need the following three lemmas.

**Lemma 6** For $k, k+1$ in $[k_{a,N}, k_{b,N}]$ we have a constant $C_0$ no depending from $k$ and $N$ such that $|\tilde{\lambda}_N^{(k+1)} - \tilde{\lambda}_N^{(k)}| \ge \frac{C_0}{N+2}$.

**Proof** : Since $f \in C^4([0, 2\pi])$, and $\rho \in C^2([\theta_1, \theta_2])$ we can write, using the main value theorem, $\tilde{\lambda}_N^{(k+1)} - \tilde{\lambda}_N^{(k)} = \frac{1}{N+2}f'(c_1) + \frac{1}{(N+2)^2}(f'\rho)'(c_2) + \frac{1}{(N+2)^3}(f'\rho'\rho)'(c_3) + \frac{1}{2}\frac{1}{(N+2)^3}(f''\rho^2)'(c_4)$ with $c_j \in ]\frac{k}{N+2}, \frac{k+1}{N+2}[$ for $j = 1, 2, 3, 4$. Then we obtain two constants $\tau_0, M_0$, depending only from $f$ and $\rho$, such that $|\tilde{\lambda}_N^{(k+1)} - \tilde{\lambda}_N^{(k)}| \ge \frac{\tau_0}{N+2} - \frac{M_0}{(N+2)^2}$. Hence if $C_0 = \frac{\tau_0}{N+2}$ we obtain the result for a sufficiently large $N$. $\square$

**Lemma 7** For $k, k+1$ in $[k_{a,N}, k_{b,N}]$ we consider the two eigenvalues $\lambda_k = \tilde{\lambda}_N^{(k)} + \frac{R_1}{(N+2)^3}$ and $\lambda_{k+1} = \tilde{\lambda}_N^{(k+1)} + \frac{R_2}{(N+2)^3}$. Then we have a constant $C_1$ no depending from $k$ such that for a sufficiently large $N$ we have $|\lambda_{k+1} - \lambda_k| \ge \frac{C_1}{N+2}$.
# 3 Appendix

## 3.1 Inversion formula for Toeplitz matrices.

We denote by $\pi_+$ and $\pi_-$ the operator defined by $\pi_+ \left( \sum_{n \in \mathbb{Z}} a_n \chi^n \right) = \sum_{n \in \mathbb{N}} a_n \chi^n$ and $\pi_- \left( \sum_{n \in \mathbb{Z}} a_n \chi^n \right) = \sum_{n \in \mathbb{N}^*} a_{-n} \chi^{-n}$.

In [19] we have state the following inversion formula

**Theorem 2** Let $P_{N+1}$ a trigonometric polynomial with degree $N+1$ and without zeros on the united disc $\bar{D}$. Let $\omega = r\bar{\chi}_0$, $0 < r < 1$, $|\chi_0| = 1$ and also $f_r = g_1, g_2$, with $g_1 = \chi_0(1 - \omega\chi)(P_{N+1})^{-1}$ and $g_2 = (1 - \omega\bar{\chi})(\overline{P_{N+1}})^{-1}$. Then for all polynomial $P$ in $\mathcal{P}_N$ we have

$$T_N(f_r)^{-1}(P) = \frac{1}{g_1}\pi_+(\frac{P}{g_2}) - \frac{1}{g_1}\pi_+\left(\Phi_N \sum_{s=0}^{+\infty} (H_{\Phi_N}^* H_{\Phi_N})^s \pi_+\left(\tilde{\Phi}_N \pi_+(\frac{P}{g_2})\right)\right).$$

with

$$\left\{ \begin{array}{ll} \tilde{\Phi}_N & = \frac{g_1}{g_2} \chi^{N+1}, \\ \tilde{\Phi}_N & = \frac{g_2}{g_1} \chi^{-(N+1)}, \\ H_{\Phi_N}(\Psi) & = \pi_-(\Phi_N \Psi), \\ H_{\Phi_N}^*(\Psi) & = \pi_+(\tilde{\Phi}_N \Psi) \end{array} \right.$$

for $\Psi \in \mathbb{H}^+$.

For the proof Theorem 2 we have to know $T_N(f_r)^{-1}$ that is also $\langle T_N(f_r)^{-1}(1)|1 \rangle$. Write $\langle T_N(f)^{-1}(1)|1 \rangle = x_0 - y_0$. The previous theorem provides

$$x_0 = \langle \pi_+ \left( \frac{1}{g_2} \right) | \frac{1}{\bar{g}_1} \rangle = \chi_0 \left| \frac{1}{P_{N+1}(0)} \right|^2.$$

To obtain $y_0$ we need the terms $\pi_+(\tilde{\Phi}_N \pi_+(\frac{1}{g_2}))$ and $\pi_+(\tilde{\Phi}_N \pi_+(\frac{1}{\bar{g}_1}))$. We have

$$\pi_+ \left( \tilde{\Phi}_N \pi_+ \left( \frac{1}{g_2} \right) \right) = \overline{P_{N+1}(0)} \pi_+ \left( \frac{g_2}{g_1} \chi^{-N-1} \right) = C_1 \frac{1}{1 - \omega \chi}$$

with

$$C_1 = \overline{P_{N+1}(0)} \bar{\chi}_0 \left( \frac{P_{N+1}(\frac{1}{\bar{\omega}})}{\overline{P_{N+1}(\bar{\omega})}} \right) \omega^{N+1} (1 - \omega^2).$$

Likewise we can write

$$\pi_+ \left( \bar{\Phi}_N \pi_+ \left( \frac{1}{\bar{g}_1} \right) \right) = C_1' \frac{1}{1 - \bar{\omega} \chi},$$

with

$$C_1' = \overline{P_{N+1}(0)} \bar{\chi}_0 \left( \frac{P_{N+1}(\frac{1}{\bar{\omega}})}{\overline{P_{N+1}(\bar{\omega})}} \right) \bar{\omega}^{N+1} (1 - \bar{\omega}^2).$$

Hence

$$y_0 = C_1 \overline{C_1'} \langle (I - H_{\Phi_N^*} H_{\Phi_N}) \frac{1}{1 - \omega \chi} | \frac{1}{1 - \bar{\omega} \chi} \rangle.$$

We have now to use the lemma
**Proof:** The property 4 implies that the rest in the equation (23) is bounded by $\frac{T}{(N+2)^3}$ where $T$ is a real no depending from $\lambda$. Hence with Lemma (6) we can write $|\lambda_{k+1} - \lambda_k| \ge \frac{C_1}{N+2} - \frac{T}{(N+2)^3}$ and with, for instance, $C_1 = \frac{C_0}{2}$ we obtain the lemma for a sufficiently large $N$. $\square$

**Lemma 8** For a fixed $k$ the equation (10) has one and only one solution in $[\theta_1, \theta_2]$.

**Proof:** Assume $\tilde{\lambda}_N$ and $\tilde{\lambda}'_N$ two solutions of (10) for a same integer $k$. By (23) and Property 4 we have $|\tilde{\lambda}_N - \tilde{\lambda}'_N| \le \frac{T'}{(N+2)^3}$) with $T'$ no depending from $\lambda$. By [1] we know that we have an eigenvalue $\lambda_{N+1}$ of the matrix $T_{N+1}(f)$ with the bound $\tilde{\lambda}_N < \lambda_{N+1} < \tilde{\lambda}'_N$ that implies $|\tilde{\lambda}_N - \lambda_{N+1}| \le \frac{T'}{(N+2)^3}$. By Lemma 6 we have $|\tilde{\lambda}_N - \lambda_{N+1}| \ge (\frac{C_1}{N+2})$, that is a contradiction with the previous estimation. $\square$

### 3 Proof of Theorem 2

First we can observe that we can define the function $H$ on $[0, \pi] \times [0, \pi]$ with $H(0,0) = f''(0)$ and $H(\pi, \pi) = f''(\pi, \pi)$. Hence for all $\lambda \in I_{0,2\pi}$ we have

$$f(\theta) - \lambda = f_1(1 - \cos \theta) - \lambda = ((1 - \cos \theta) - (1 - \cos \theta_{\lambda})) H_{\lambda}(\theta)$$

where $H_\lambda: \theta \mapsto H(\theta, \theta_\lambda)$ is a regular function on $[-\pi, \pi]$. for all $\lambda \in [0, \pi]$. With the same notations as in the proof of Theorem 1 we can still write, for $\lambda \notin \text{Spec}(T_N(f))$

$$T_{1,N,\lambda} = \frac{\det (T_{N-1}(f) - \lambda I_{N-1})}{\det (T_N(f) - \lambda I_N)}.$$

As previously we have also

$$\lambda \in \text{Spec}(T_N(f)) \iff \frac{\det (T_N(f) - \lambda I_{N-1})}{\det (T_{N-1}(f) - \lambda I_N)} = 0.$$

and, always with the equation (28), we can write

$$\frac{\det (T_N(f) - \lambda I_{N-1})}{\det (T_{N-1}(f) - \lambda I_N)} = \frac{1 - \bar{\chi}_\lambda^{2(N+1)} \tau_N(\chi_\lambda)}{(1 - \bar{\chi}_\lambda^{2(N+2)} \tau_N(\chi_\lambda)) B_{2,N,\lambda} - B_{1,N,\lambda}}, \quad (24)$$

with

$$\tau_N(\theta_\lambda) = \frac{\bar{P}_{N+1,\lambda}(\chi_\lambda)P_{N+1,\lambda}(\chi_\lambda)}{\bar{P}_{N+1,\lambda}(\bar{\chi}_\lambda)P_{N+1,\lambda}(\bar{\chi}_\lambda)},$$

and $B_{1,N,\lambda}, B_{2,N,\lambda}$ as previously. Hence we can write

$$\lambda \in (\lambda \in \text{Spec}(T_N(f))) \cap I_{0,\pi} \iff \chi_{\lambda}^{2(N+2)} = \tau_N(\theta_{\lambda}), \lambda \in I_{0,\pi}. \quad (25)$$

Since the function $H_\lambda$ is even, the constant $\tau_N(\theta_\lambda)$ can be rewritten as

$$\tau_N(\theta_\lambda) = \left( \frac{P_{N+1,\lambda}(\chi_\lambda)}{P_{N+1,\lambda}(\bar{\chi}_\lambda)} \right)^2.$$
**Lemma 9** $\frac{1}{1-\omega\chi}$ is an eigenvector of $H_{\Phi_N}^* H_{\Phi_N}$ for the eigenvalue $\tau_{N,r}(\omega)\omega^{2(N+2)}$ with $\tau_{N,r}(\omega) = \frac{\bar{P}_{N+1}(\frac{1}{\omega})P_{N+1}(\frac{1}{\omega})}{\bar{P}_{N+1}(\omega)P_{N+1}(\omega)}$, with $|\tau_{N,r}(\omega)| = 1$ for $0 < r < 1$.

It is the lemma 1 of [19]. We obtain

$$y_0 = C_1 \overline{C_1'} \frac{1}{1 - \omega^{2N+2} \tau_{N,r}(\omega)} \frac{1}{1 - \omega^2}.$$

If now we consider the function $f_1$ defined by the product $f_1 = \tilde{g}_1\tilde{g}_2$ with $\tilde{g}_1 = \chi_0(1-\bar{\chi}_0\chi)\frac{1}{\bar{P}_{N+1}}$ and $\tilde{g}_2 = (1-\bar{\chi}_0\bar{\chi})\frac{1}{\bar{P}_{N+1}}$, then for a fixed $N$ $\lim_{r \to 1} (T_N f_r)^{-1}_{1,1} = (T_N f)^{-1}_{1,1}$. Indeed

$$(T_N f_r)^{-1} (T_N f) = (T_N f_r)^{-1} (T_N f_r) + (T_N f_r)^{-1} (T_N (f_1 - f_r)).$$

And $\lim_{r \to 1} (T_N(f - f_r)) = 0$ that implies $\lim_{r \to 1} (T_N f_r)^{-1} (T_N f) = I_N$. Hence we can conclude that

$$(T_N(f))^{-1}_{1,1} = \frac{\left(1 - \chi_0^{2(N+2)} \tau_N(\chi_0)\right) B_{2,N} - B_{1,N}}{1 - \chi_0^{2(N+2)} \tau_N},$$

with $B_{1,N} = C_1 \overline{C_1'}(1 - \bar{\chi}_0^2)$, $B_{2,N} = \chi_0 \left| \frac{1}{\bar{P}_{N+1}(0)} \right|^2$, and $\tau_N(\chi_0) = \frac{\overline{P_{N+1}(\bar{\chi}_0)P_{N+1}(\bar{\chi}_0)}}{\overline{P_{N+1}(\chi_0)P_{N+1}(\chi_0)}}$

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[8] A. B\"ottcher and B. Silbermann. *Introduction to large truncated Toeplitz matrices*. Springer Verlag, 1999.

[9] R. Dahlhaus. Efficient parameter estimation for self-similar processes. *Ann. Statist.*, 17:1749-1766, 1989.
On the other hand the function $\theta \mapsto \frac{P_{N+1,f(\theta)}(\bar{\chi}_{\lambda})}{P_{N+1,f(\theta)}(\chi_{\lambda})}$ is continuous from $[0, 2\pi]$ to $\{z||z|=1\}$ hence we have a function $\rho_N$ defined and continuous on $[0, \pi]$ such that $\tau_N(\theta_\lambda) = e^{2i\rho_N(\theta_\lambda)}$. Then equation (25) can be written

$$ \lambda \in (\lambda \in \text{Spec}(T_N(f))) \cap I_{0,\pi} \iff \theta_\lambda = \frac{\rho_N(\theta_\lambda) + k\pi}{(N+2)}, k \in [0, 2N+3]. \quad (26) $$

Hence for $k$ in $\{0, \dots, 2N+3\}$ we have to find the solution in $[0, \pi]$ of the equation

$$ (N+2)\theta - \rho_N(\theta) = k\pi \quad (27) $$

But in the particular case where $\theta_1 = 0$ and $\theta_2 = \pi$ it is easy to verify that the function $\rho_N$ is in fact an odd $2\pi$-periodic function with $\rho_N(0) = 0$. Now we denote by $F_N$ the function $\theta \mapsto (N+2)\theta - \rho_N(\theta)$. For $1 \le k \le N+1$ we have

$$ F_N(0) = 0 < \pi k, F_N(\pi) = (N+2)\pi > \pi k. $$

Hence the equation (27) has at least one solution in $[0, \pi]$ for all $k \in \{0, \dots, N+1\}$. In the other hand it is obvious that the solution of the equations $\theta = \frac{\rho_N(\theta)+k\pi}{(N+2)}$ and $\theta = \frac{\rho_N(\theta)+k'\pi}{(N+2)}$ are different for $k \ne k'$. Since $f$ is strictly increasing on $[0, \pi]$, we have found $N+1$ eigenvalues of $T_N(f)$ in $[f(0), f(\pi)]$, and we will not obtain other eigenvalues outside the set $\{0, \dots, N+1\}$. The rest of the proof is the same as the proof of Theorem 1.

# 4 Appendix

For the proof of Theorem 1 we have to know $T_N(f)^{-1}_{1,1}$. First we use Theorem 3 to obtain $T_N(f_r)^{-1}_{1,1}$ with $f_r = \chi_\lambda(1 - r\bar{\chi}_\lambda\bar{\chi})(1 - r\bar{\chi}_\lambda\bar{\chi})\frac{1}{|P_{N+1,\lambda}|^2}$, and now $g_1 = \chi_\lambda(1 - r\bar{\chi}_\lambda\bar{\chi})\frac{1}{P_{N+1,\lambda}}$, $g_2 = (1 - r\bar{\chi}_\lambda\bar{\chi})\frac{1}{P_{N+1,\lambda}}$

We have to observe that $T_N(f_r)^{-1}_{1,1}$ that is also $\langle T_N(f_r)^{-1}(1)|1\rangle$. Write $\langle T_N(f_r)^{-1}(1)|1\rangle = x_0 - y_0$. Theorem 3 provides

$$ x_0 = \langle \pi_+ \left( \frac{1}{g_2} \right) | \frac{1}{\bar{g}_1} \rangle = \frac{\chi_\lambda}{|P_{N+1,\lambda}(0)|^2}. $$

To obtain $y_0$ we need the terms $\pi_+ (\tilde{\Phi}_N \pi_+ (\frac{1}{g_2}))$ and $\pi_+ (\bar{\Phi}_N \pi_+ (\frac{1}{\bar{g}_1}))$. We have, if $\omega = r\bar{\chi}_\lambda$,

$$ \pi_+ \left( \tilde{\Phi}_N \pi_+ \left( \frac{1}{g_2} \right) \right) = \overline{P_{N+1,\lambda}(0)} \pi_+ \left( \frac{g_2}{g_1} \chi^{-N-1} \right) = C_1 \frac{1}{1 - \omega \chi} $$

with

$$ C_1 = \overline{P_{N+1,\lambda}(0)} \bar{\chi}_\lambda \left( \frac{P_{N+1,\lambda}(\frac{1}{\bar{\omega}})}{\bar{P}_{N+1}(\bar{\omega})} \right) \omega^{N+1} (1 - \omega^2). $$

Likewise we can write

$$ \pi_+ \left( \bar{\Phi}_N \pi_+ \left( \frac{1}{\bar{g}_1} \right) \right) = C'_1 \frac{1}{1 - \bar{\omega} \chi}, $$
[10] P. Deift, A. Its, and I. Krasovsky. Toeplitz matrices and Toeplitz determinants under the impetus of the ising model. some history and some recent results., *Comm. Pure Appl. Math.*, 66(1360-1438.), 2013.

[11] S.E. Ekström, I. Furci, and S. Serra-Capizzano. Exact formulae and matrix-less eigen-solvers for block banded symmetric Toeplitz matrices. *BIT Numerical Mathematics*, 58(4):937–968, December 2018.

[12] S.E. Ekström and S. Serra-Capizzano. Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols. *Numer. Linear Alg with Appl*, 25(5), October 2018.

[13] U. Grenander and G. Szegö. *Toeplitz forms and their applications*. Chelsea, New York, Second edition, 1984.

[14] Y. Haugazeau. Application du théorème de sylvester à la localisation des valeurs propres $ax = λbx$ dans le cas symétrique. *RAIRO. Analyse numérique*, 14(1):25-41, 1980.

[15] H.J. Landau. Maximum entropy and the moment problem. *Bull. Amer. Math. Soc.*, 16(1):47-77, 1987.

[16] S. Parter. Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations. *Trans. Amer. Math. Soc.*, 99:153-192, 1961.

[17] S. Parter. On the extreme eigenvalues of Toeplitz matrices. *Trans. Amer. Math. Soc.*, 100:263-270, 1961.

[18] S. Parter. On the extreme eigenvalues of truncated Toeplitz matrices. *Bull. Amer. Math. Soc.*, 67:191-196, 1961.

[19] P. Rambour and A. Seghier. Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz. *Bull. des Sci. Math.*, 131:258-275, 2006.

[20] Jean-Marc Rinkel. Inverses et propriétés spectrales des matrices de Toeplitz à symbole singulier. *Ann. Fac. Sci. Toulouse Math.*, 11(1):71-103, 2002.

[21] S. Serra. On the extreme eigenvalues of Hermitian (block) Toeplitz matrices. *Linear Algebra Appl.*, 270:109-129, 1998.

[22] W.F. Trench. Interlacement of the even and odd spectra of real symmetric Toeplitz matrices. *Linear Alg. Appl.*, 195:59-68, 1993.

[23] W.F. Trench. Asymptotic distribution of the spectra of a class of generalized Kac-Murdoch-Szegö matrices. *Linear Algebra Appl.*, 294:181-192, 1999.

[24] E.E. Tyrtyshnikov and N.L. Zamarashkin. Distribution of the eigenvalues and singular numbers of toeplitz matrices under weakened requirements on the generating function., *Sb. Maths.*, 48:1191-1201, 1997.

[25] E.E. Tyrtyshnikov and N.L. Zamarashkin. Toeplitz eigenvalues for Radon measures., *Linear Algebra Appl.*, 343/344:345-354., 2002.

[26] H. Widom. On the eigenvalues of certain hermitian operators. *Trans. Amer. Math. Soc.*, 88:491-522, 1958.
with

$$C'_{1} = \overline{P_{N+1,\lambda}(0)\bar{\chi}_{\lambda}} \left( \frac{P_{N+1,\lambda}(\frac{1}{\bar{\omega}})}{\overline{P_{N+1,\lambda}(\bar{\omega})}} \right) \bar{\omega}^{N+1}(1 - \bar{\omega}^2).$$

Hence

$$y_0 = C_1 \overline{C'_1} \langle (I - H_{\Phi_N^*} H_{\Phi_N})^{-1} \frac{1}{1 - \omega\chi} | \frac{1}{1 - \bar{\omega}\chi} \rangle.$$

We have now to use the following lemma

**Lemma 9** $\frac{1}{1-\omega\bar{\chi}}$ is an eigenvector of $H_{\Phi_N}^* H_{\Phi_N}$ for the eigenvalue $\tau_{N,r}(\omega)\omega^{2(N+2)}$ with $\tau_{N,r}(\omega) = \frac{\overline{P_{N+1,\lambda}(\frac{1}{\bar{\omega}})P_{N+1,\lambda}(\frac{1}{\bar{\omega}})}}{\overline{P_{N+1,\lambda}(\omega)P_{N+1,\lambda}(\omega)}}$, with $|\omega^{2(N+2)}\tau_{N,r}(\omega)| < 1$ for $r \to 1$ and $N$ sufficiently large.

It is Lemma 1 of [24]. We obtain

$$y_0 = C_1 \overline{C'_1} \frac{1}{1 - \omega^{2N+2}\tau_{N,r}(\omega)} \frac{1}{1 - \omega^2}.$$

If now we consider the function $f_1$ defined by the product $f_1 = \tilde{g}_1\tilde{g}_2$ with $\tilde{g}_1 = \chi_0(1-\bar{\chi}_0\chi)\frac{1}{\overline{P_{N+1}}}$ and $\tilde{g}_2 = (1-\bar{\chi}_0\bar{\chi})\frac{1}{\overline{P_{N+1}}}$, then for a fixed $N$, $\lim_{r \to 1} (T_N f_r)^{-1}_{1,1} = (T_N f)^{-1}_{1,1}$. Indeed

$$(T_N f_r)^{-1} (T_N f) = (T_N f_r)^{-1} (T_N f_r) + (T_N f_r)^{-1} (T_N (f_1 - f_r)).$$

And $\lim_{r \to 1} (T_N(f - f_r)) = 0$ that implies $\lim_{r \to 1} (T_N f_r)^{-1} (T_N f) = I_N$. Hence we can conclude that

$$((T_N(f))^{-1})_{1,1} = \frac{\left(1 - \bar{\chi}_\lambda^{2(N+2)} \tau_N(\chi_\lambda)\right) B_{2,N,\lambda} - B_{1,N,\lambda}}{1 - \bar{\chi}_\lambda^{2(N+2)} \tau_N(\chi_\lambda)}, \quad (28)$$

with $B_{1,N,\lambda} = C_1 \overline{C'_1}(1 - \bar{\chi}_\lambda^2)^{-1}$, $B_{2,N,\lambda} = \frac{\chi_\lambda}{|P_{N+1,\lambda}(0)|^2}$, and $\tau_N(\chi_\lambda) = \frac{\overline{P_{N+1,\lambda}(\chi_\lambda)P_{N+1,\lambda}(\chi_\lambda)}}{\overline{P_{N+1,\lambda}(\bar{\chi}_\lambda)P_{N+1,\lambda}(\bar{\chi}_\lambda)}}$

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