# Lower bounds in $H^2$-rational approximation to $z^N$

Laurent Baratchart, Sylvain Chevillard

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Laurent Baratchart, Sylvain Chevillard. Lower bounds in $H^2$-rational approximation to $z^N$. 2025. hal-05385529

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Distributed under a Creative Commons Attribution 4.0 International License
Lower bounds in $H^2$-rational approximation to $z^N$.

Laurent Baratchart†

Sylvain Chevillard†

Laurent.Baratchart@sophia.inria.fr, phone: +33 4 92 38 78 74, fax: +33 4 92 38 78 58.

sylvain.chevillard@inria.fr, phone: +33 4 92 38 76 42, fax: +33 4 92 38 78 58.

January 2025

**Abstract:** We derive lower bounds in best rational approximation of given degree $n \le N$ to $z^N$ in the Hardy space $H^2$ of the unit disk.

**Keywords:** Complex rational approximation, Hardy spaces, lower bounds, error rates.

*Classification numbers (AMS):* 31B05, 35J25, 42B35, 46E20, 47B35.

**Acknowledgement** The authors thank A. François, from the INRIA-team SIERRA joint with CNRS (DI ENS, UMR 8548), for bringing this problem to their attention.

Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Notations and Preliminaries</td><td>2</td></tr><tr><td>3 Hardy spaces of the disk</td><td>4</td></tr><tr><td>4 Best rational and meromorphic approximation on the circle</td><td>6</td></tr><tr><td>5 $H^2$ rational approximation to $z^N$: lower bounds</td><td>9</td></tr></table>

<footnote n="2">²Inria, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France.</footnote>
# 1 Introduction

Rational approximation on the circle or the line to functions with constant modulus is a classical issue relating to model reduction of linear delay control systems. In recent years, the subject was somewhat revived by the rise of black box modeling using neural nets.

It is well known that rational approximation of lower degree, in the sup norm, to a function of constant modulus on the circle or the line extending holomorphically to the disk or the half-plane is ineffective: the best approximant is 0. The situation is different for $L^2$ approximants, though one may surmise that rational approximation of lower degree, though not completely ineffective, should be slow in this case. This has been substantiated in [5] where lower bounds in rational approximation to Blaschke products (of finite or infinite degree) were derived in terms of their zeros. However, lower bounds that involve comparing the degree of the approximated function (assumed rational of constant modulus) with the degree of the approximant (assumed of course to be smaller) were apparently not obtained so far.

In Theorem 2 of the present paper, we present such a result in the special case where the approximated function is of the form $z^N$. This is the transfer function of a pure delay system in discrete time. The proof is an interesting modification of the standard argument, valid for the sup norm, making use of the degree formula for unimodular functions in terms of the fractional Sobolev $W^{1/2}$-norm due to H. Brézis. It is to be hoped that suitable refinements thereof will produce lower bounds that are valid for more general Blaschke products than $z^N$.

# 2 Notations and Preliminaries

Let $\mathbb{D}$ be the unit disk and $\mathbb{T}$ the unit circle in the complex plane $\mathbb{C}$. We put $z = x + iy$ to denote the real and imaginary parts of $z \in \mathbb{C}$, which makes for the identification $\mathbb{C} \sim \mathbb{R}^2$.

We write $P_n[z]$ for the space of complex algebraic polynomials of degree at most $n$ in the variable $z$, or simply $P_n$ if the variable is understood. Below, we let $\mathcal{Z}(q)$ indicate the set of zeros of a polynomial $q$. For $q_n \in P_n[z]$, we define its reciprocal polynomial to be

$$ \tilde{q}_n(z) := z^n \overline{q_n(1/\bar{z})}. \qquad (1) $$

One should be careful that this definition depends on $n$: if we consider $q \in P_{n-1}$ as an element of $P_n$ with zero leading coefficient, the two definitions of $\tilde{q}(z)$ in $P_{n-1}$ and $P_n$ may be inconsistent. Therefore we always specify which definition is used; e.g., via a subscript “n” for $q_n$ like we did in (1). Clearly the “tilde” operation is an involution of $P_n$ preserving the pointwise modulus on $\mathbb{T}$.

We designate by $\mathcal{R}_{m,n} = \mathcal{R}_{m,n}(z)$ the set of complex rational functions of type $(m,n)$ in $L^2$, namely those that can be written as $p_m/q_n$ where $p_m$ belongs to $P_m$ and $q_n \in P_n$ has no root on $\mathbb{T}$. When $r = p_m/q_n$ is in irreducible form, the integer $\max\{m,n\}$ is the (exact) degree of $r$.

We denote by $C(\mathbb{T})$ the space of continuous, complex-valued functions on $\mathbb{T}$, and by $C^\infty(\mathbb{T})$ the subspace of smooth functions; that is, functions $f(e^{i\theta})$ that are locally $C^\infty$-smooth functions of $\theta \in \mathbb{R}/(2\pi\mathbb{Z})$. Hereafter, for a function on $\mathbb{T}$, a superscript “prime” always refers to the derivative with respect to $\theta$. Note that $f'$ is indeed well-defined, because $\frac{d}{d\theta}f(e^{i\theta})$ is independent of the continuous branch $\theta$ of the argument which is chosen modulo $2\pi$. For $1 \le p \le \infty$, we let $L^p = L^p(\mathbb{T})$ indicate the familiar Lebesgue space of (equivalence classes of a.e. coinciding) complex measurable functions on $\mathbb{T}$ such that

$$ \|f\|_p = \left( \frac{1}{2\pi} \int_0^{2\pi} |f(e^{i\theta})|^p d\theta \right)^{1/p} < \infty \quad \text{if } 1 \le p < \infty, \qquad \|f\|_\infty = \text{ess. sup}_{\theta \in [0,2\pi]} |f(e^{i\theta})| < \infty. $$

The definition extends to $\mathbb{C}^n$-valued functions, replacing moduli by Euclidean norms. In a similar vein, we put $C(\mathbb{D})$ and $C^\infty(\mathbb{D})$ for continuous and smooth functions on $\mathbb{D}$. We denote by $C(\bar{\mathbb{D}})$ (resp. $C^\infty(\bar{\mathbb{D}})$) those extending continuously to $\bar{\mathbb{D}}$ (resp. smoothly to a neighborhood of $\bar{\mathbb{D}}$). The Lebesgue spaces $L^p(\mathbb{D})$ are defined
analogously, namely:

$$ \|f\|_{L^p(\mathbb{D})} = \left( \frac{1}{\pi} \int_{\mathbb{D}} |f(z)|^p dm(z) \right)^{1/p} < \infty \quad \text{if } 1 \le p < \infty, \quad \|f\|_{L^\infty(\mathbb{D})} = \operatorname{ess. sup}_{z \in \mathbb{D}} |f(z)| < \infty, $$

where $m$ indicates Lebesgue measure.

The Sobolev space $W^{1,2}$ consists of complex functions on $\mathbb{T}$, absolutely continuous with respect to arclength, whose derivative $f'(e^{i\theta}) := \frac{d}{d\theta}f(e^{i\theta})$ again lies in $L^2$. It is a Banach space with norm

$$ \|f\|_{W^{1,2}}^2 = \|f\|_{L^2}^2 + \|f'\|_{L^2}^2. $$

If one associates with $f: \mathbb{T} \to \mathbb{C}$ its Fourier coefficients $(a_k)_{k \in \mathbb{Z}}$, so that $f(e^{i\theta}) = \sum_{k \in \mathbb{Z}} a_k e^{ik\theta}$ is the Fourier expansion of $f$, one may identify $L^2$ with $l^2(\mathbb{Z})$ and $W^{1,2}$ with the weighted space $l^2(\mathbb{Z}, w)$, where the weight is $w(k) = 1 + |k|^2$; moreover this identification is isometric, because by Parseval's relation: one has

$$ \|f\|_2 = \left( \sum_{k \in \mathbb{Z}} |a_k|^2 \right)^{1/2} \quad \text{and} \quad \|f'\|_2 = \left( \sum_{k \in \mathbb{Z}} |k|^2 |a_k|^2 \right)^{1/2}. $$

The fractional Sobolev space $W^{1/2,2}$ is a real interpolation space of exponent $1/2$ between $L^2$ and $W^{1,2}$, a norm on which is given in terms of the Fourier coefficients (see [2, thm 7.55]) by

$$ \|f\|_{1/2}^2 = |a_0|^2 + \sum_{k=-\infty}^{\infty} |k||a_k|^2, \quad f(e^{i\theta}) = \sum_{k=-\infty}^{\infty} a_k e^{ik\theta}. \quad (2) $$

Though the norm in (2) is the one we shall use, let us mention that an equivalent norm on $W^{1/2,2}$ is given in terms of the values of $f$ by

$$ \|f\|_{W^{1/2,2}} := \|f\|_{L^2} + \left( \int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|f(e^{i\theta}) - f(e^{i\theta'})|^2}{|e^{i\theta} - e^{i\theta'}|^2} d\theta d\theta' \right)^{1/2}, \quad (3) $$

see [2, thm 7.47 & rem. 7.45]. In fact, the norm in (3) characterizes $W^{1/2,2}$ as the space of traces on $\mathbb{T}$ of Sobolev functions in $W^{1,2}(\mathbb{D})$; the latter space consists of complex functions in $L^2(\mathbb{D})$ whose distributional derivatives of the first order again lie in $L^2(\mathbb{D})$. We refer the reader to [2] for more on Sobolev spaces.

The winding number $W(f)$ of a nowhere vanishing $f \in C(\mathbb{T})$ is the number of times the tip of the vector $f(e^{i\theta})$ winds around the origin as $\theta$ traverses $[0, 2\pi)$. When $f \in C^\infty(\mathbb{T})$, so that $f = \rho e^{i\alpha}$ for some smooth functions $\rho: \mathbb{T} \to \mathbb{R}^+$ and $\alpha: \mathbb{T} \to \mathbb{R}/(2\pi\mathbb{Z})$ with $\rho > 0$, the winding number may formally be defined as

$$ W(f) := \frac{1}{2\pi} \int_{0}^{2\pi} (\alpha(e^{i\theta}))' d\theta = \frac{1}{2i\pi} \int_{0}^{2\pi} \frac{(e^{i\alpha})'}{e^{i\alpha}}(e^{i\theta}) d\theta = \frac{1}{2i\pi} \int_{0}^{2\pi} \frac{f'}{f}(e^{i\theta}) d\theta, \quad (4) $$

where we used that

$$ \int_{0}^{2\pi} \frac{(\rho(e^{i\theta}))'}{\rho(e^{i\theta})} d\theta = [\log \rho(e^{i\theta})]_{\theta=0}^{\theta=2\pi} = 0. $$

When $f$ is merely continuous and never vanishing, for each $\varepsilon > 0$ one can pick a smooth function $g \in C^\infty(\mathbb{T})$ such that $|f - g| < \varepsilon|f|$ on $\mathbb{T}$, and for $\varepsilon$ small enough any two such $g$ are smoothly homotopic in a never vanishing manner, hence they have the same winding number because the latter is integer-valued and varies continuously along the homotopy. This common winding number is, by definition, the one of $f$. In particular, if $f \in C(\mathbb{D})$ is never zero on $\mathbb{T}$ and meromorphic in $\mathbb{D}$, then

$$ W(f|_{\mathbb{T}}) = \sharp Z(f) - \sharp P(f) \quad (5) $$
where $\#Z(f)$ is the number of zeros and $\#P(f)$ the number of poles of $f$ in $\mathbb{D}$, counting multiplicities. Indeed, $f$ can be uniformly approximated by $f_r(e^{i\theta}) := f(re^{i\theta})$ for $r$ small enough, and then (5) follows from (4) and the residue formula.

Let us stress that the winding number is a special case of the more general notion of topological degree, that applies in higher dimension as well. More precisely, for $F: \mathbb{S}^n \to \mathbb{S}^n$ a $C^\infty$-smooth map from the unit sphere of dimension $n$ into itself (viewed here as a smooth oriented manifold), the topological degree of $F$ is

$$ \deg F := \sum_{\xi \in F^{-1}(c)} \mathrm{orient}(DF(\xi)), \qquad (6) $$

where $c$ is any regular value of $F$ (a value is regular if it is not attained at a point $\xi$ where the rank of the derivative $DF(\xi)$ drops) and $\mathrm{orient}(DF(\xi))$ is equal to +1 if $DF(\xi)$ preserves orientation and to -1 if it reverses orientation. Note that the right hand side of (6) is a well-defined finite sum because $c$ is a regular value. The topological degree of a continuous map $F$ is then defined as the one of a smooth map arbitrary close to $F$. When $n = 1$ and $\mathbb{S}^1 \sim \mathbb{T}$ is oriented counterclockwise while $F$ is smooth so that $F = e^{i\alpha}$ for some smooth $\alpha: \mathbb{T} \to \mathbb{R}/(2\pi\mathbb{Z})$, a regular value is a direction $e^{it_0}$ which is attained under $F$ only at points $e^{i\theta}$ where $(\alpha(e^{i\theta}))' \neq 0$, and $\mathrm{orient}(DF(e^{i\theta})) = +1$ if $F(e^{i\theta})$ traverses $e^{it_0}$ counterclockwise $((\alpha(e^{i\theta}))' > 0)$ and $-1$ if it traverses clockwise $((\alpha(e^{i\theta}))' < 0)$. It is thus apparent that deg $F$ algebraically counts the number of times $F(e^{i\theta})$ winds around zero as $\theta$ percurses $[0, 2\pi)$. Hence, the winding number of a continuous never-vanishing $f: \mathbb{T} \to \mathbb{C}$ is just the topological degree of $\frac{f}{|f|}: \mathbb{T} \to \mathbb{T}$. We refer the reader to [10, Chapter 3] for basic facts on the topological degree (see in particular the second corollary on p. 115 of that reference), and its computation as a sum of local indices at the singular points of any nondegenerate extension of $F$ inside the ball that generalizes (5) (see proposition on page 108 and the computation in the proof of proposition on p. 110 of [10]).

When $f: \mathbb{T} \to \mathbb{T}$ is continuous and lies in $W^{1/2,2}$ (membership in $W^{1/2,2}$ alone does not imply continuity), a remarkable formula of Brézis asserts that

$$ W(f) = \sum_{k \in \mathbb{Z}} k|a_k|^2, \qquad f(e^{i\theta}) = \sum_{k=-\infty}^{\infty} a_k e^{ik\theta}; \qquad (7) $$

see [7] for details. In this case, it was later shown by Bourgain and Kahane that if $W(f) = 0$ then

$$ \sum_{k \in \mathbb{Z}} |k| |a_k|^2 \le 32 \sum_{k>0} k |a_k|^2, \qquad (8) $$

see [6, Theorem 1 & Equation (25)].

## 3 Hardy spaces of the disk

We let $H^2 = H^2(\mathbb{D})$ be the Hardy space of holomorphic functions in $\mathbb{D}$ whose Taylor coefficients at 0 are square summable:

$$ H^2 = \{f(z) = \sum_{k=0}^{\infty} a_k z^k : \|f\|_{H^2} := \sum_{k=0}^{\infty} |a_k|^2 < +\infty\}. $$

We refer the reader to [9] for standard facts on Hardy spaces. By Parseval's relation

$$ \|f\|_{H^2}^2 = \sup_{0 \le r < 1} \frac{1}{2\pi} \int_0^{2\pi} |f(re^{i\theta})|^2 d\theta, \qquad (9) $$

and the map

$$ (f(z) = \sum_{k=0}^{\infty} a_k z^k) \to (f^*(e^{i\theta}) := \sum_{k=0}^{\infty} a_k e^{ik\theta}) $$

is an isometry from $H^2$ onto the closed subspace of $L^2$ comprised of functions whose Fourier coefficients of strictly negative index do vanish. As is customary, we identify $H^2$ with this subspace so that the distinction
between $f$ and $f^*$ as well as $\|f\|_{H^2}$ and $\|f^*\|_2$ will disappear. This conveniently allows one to regard members of the Hardy class both as functions on $\mathbb{D}$ and on $\mathbb{T}$. From the function-theoretic viewpoint, the correspondence $f \mapsto f^*$ is that $f^*(e^{i\theta})$ is almost everywhere the limit of $f(z)$ as $z$ tends non-tangentially to $e^{i\theta}$ within $\mathbb{D}$.

We put $\bar{H}^{2,0} = \bar{H}^{2,0}(\mathbb{C} \setminus \bar{\mathbb{D}})$ for the companion Hardy space of holomorphic functions in $\mathbb{C} \setminus \bar{\mathbb{D}}$, vanishing at infinity, whose Taylor coefficients there are square summable:

$$ \bar{H}^{2,0} = \{ f(z) = \sum_{k=1}^{\infty} a_k z^{-k} : \|f\|_{\bar{H}^{2,0}} := \sum_{k=1}^{\infty} |a_k|^2 < +\infty \}. $$

The map

$$ (f(z) = \sum_{k=1}^{\infty} a_k z^{-k}) \to (f^*(e^{i\theta}) = \sum_{k=1}^{\infty} a_k e^{-ik\theta}) $$

is an isometry from $\bar{H}^{2,0}$ onto the closed subspace of $L^2$ comprised of functions whose Fourier coefficients of non-negative index do vanish, and as before we identify $\bar{H}^{2,0}$ with the latter. It holds the orthogonal sum:

$$ L^2 = H^2 \oplus \bar{H}^{2,0}. \qquad (10) $$

Clearly, $f$ belongs to $\bar{H}^{2,0}$ if and only if the function $\tilde{f}$ given by

$$ \tilde{f}(z) := z^{-1} \overline{f(1/z)} \qquad (11) $$

lies in $H^2$, and the map $f \mapsto \tilde{f}$ is an involutive isometry of $L^2$ sending $H^2$ onto $\bar{H}^{2,0}$. Actually $|\tilde{f}| = |f|$ pointwise on $\mathbb{T}$, since $f(1/\bar{z}) = f(z)$ when $|z| = 1$. If $f$ is holomorphic on $\Omega$, then $f^*(z) = f(1/\bar{z})$ is holomorphic on the reflection of $\Omega$ across $\mathbb{T}$, and if $f$ is rational $f^*$ is likewise rational of the same degree. By inspection, we get that $f_{|\mathbb{T}}^\sharp = \bar{f}_{|\mathbb{T}}$ where the subscript $|\mathbb{T}$ indicates “restriction on $\mathbb{T}$”. We let

$$ \mathbf{P}_+(\sum_{k \in \mathbb{Z}} a_k e^{ik\theta}) = \sum_{k \ge 0} a_k e^{ik\theta} \quad \text{and} \quad \mathbf{P}_-(\sum_{k \in \mathbb{Z}} a_k e^{ik\theta}) = \sum_{k < 0} a_k e^{ik\theta} \qquad (12) $$

denote the so-called Riesz projections, that discard the Fourier coefficients of strictly negative and non-negative index respectively. Clearly $\mathbf{P}_+$ (resp. $\mathbf{P}_-$) contractively maps $L^2$ onto $H^2$ (resp. $\bar{H}^{2,0}$) and $\mathbf{P}_+ + \mathbf{P}_- = I$. Note that, by Cauchy's formula, $\mathbf{P}_\pm(f)$ can be expressed as a Cauchy integral:

$$ \mathbf{P}_+(f)(z) = \frac{1}{2i\pi} \int_{\mathbb{T}} \frac{f(\zeta)}{\zeta - z} d\zeta, \quad |z| < 1, \qquad \mathbf{P}_-(f)(z) = \frac{1}{2i\pi} \int_{\mathbb{T}} \frac{f(\zeta)}{z - \zeta} d\zeta, \quad |z| > 1. \qquad (13) $$

The Hardy space $H^\infty = H^\infty(\mathbb{D})$ consists of bounded holomorphic functions on $\mathbb{D}$, endowed with the sup norm. From (9) we see that $H^\infty$ embeds contractively in $H^2$, in particular each $f \in H^\infty$ has a non-tangential limit $f^*$ on $\mathbb{T}$. It can be shown that $\|f^*\|_\infty = \|f\|_{H^\infty}$, and that the map $f \mapsto f^*$ is an isometry from $H^\infty$ onto the closed subspace of $L^\infty$ comprised of functions whose Fourier coefficients of strictly negative index do vanish. Again we identify $H^\infty$ with this subspace. Likewise, the space $\bar{H}^{\infty,0}$ of bounded holomorphic functions vanishing at infinity in $\mathbb{C} \setminus \bar{\mathbb{D}}$ identifies via non-tangential limits with the closed subspace of $L^\infty$ consisting of functions whose Fourier coefficients of non-negative index do vanish. However, in contrast with the situation for $L^2$, the operators $\mathbf{P}_\pm$ are unbounded on $L^\infty$.

A fundamental fact [9, ch. II, cor. 5.7] is that each nonzero $f \in H^2$ factors uniquely as $f = jw$ where

$$ w(z) = \exp \left\{ \frac{1}{2\pi} \int_{0}^{2\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \log |f(e^{i\theta})| d\theta \right\} \qquad (14) $$

belongs to $H^2$ and is called the *outer factor* of $f$, normalized to be positive at zero, while $j \in H^\infty$ has modulus 1 a.e. on $\mathbb{T}$ and is called the *inner factor* of $f$. The latter decomposes further as $j = bS$, where

$$ b(z) = cz^k \prod_{\zeta_l \neq 0} \frac{-\bar{\zeta}_l}{|\zeta_l|} \frac{z - \zeta_l}{1 - \bar{\zeta}_l z} \qquad (15) $$
is the Blaschke product, with zero of multiplicity $k \ge 0$ at the origin, associated to a sequence of points $\zeta_l \in \mathbb{D} \setminus \{0\}$ and to a constant $c \in \mathbb{T}$, while

$$S(z) = \exp \left\{ -\frac{1}{2\pi} \int_{0}^{2\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} d\mu(\theta) \right\}$$

is the singular inner factor associated with a positive singular measure $\mu$ on $\mathbb{T}$. The $\zeta_l$ are of course the zeros of $f$ in $\mathbb{D} \setminus \{0\}$, counting multiplicities by repetition. The total number of zeros, finite or infinite, is called the degree of the Blaschke product. When the latter is finite, the Blaschke product is rational and the two notions of degree that we introduced do coincide. Throughout, we let $B_n$ denote the set of Blaschke products of degree at most $n$. When $m < \infty$, note that $B_m \subset \mathcal{R}_{m,m}$ is comprised of rational functions of degree at most $m$ which are analytic in $\mathbb{D}$ and have unit modulus everywhere on $\mathbb{T}$. Alternatively, $B_m$ consists of functions of the form $q_k/\bar{q}_k$ where $q_k \in \mathcal{P}_k$ has exact degree $k \le m$ and all its roots in $\mathbb{D}$. Clearly, $B_m$ is included in the unit sphere of both $H^2$ and $H^\infty$. If the degree is infinite, the convergence of the product in (15) is equivalent to the condition

$$\sum_l (1 - |\zeta_l|) < \infty \qquad (16)$$

which holds automatically when $f \in H^2$. That $w(z)$ is well-defined rests on the fact that $\log|f| \in L^1$ if $f \in H^2 \setminus \{0\}$. A function $f \in H^2$ with inner-outer factorization $f = jw$ lies in $H^\infty$ if, and only if $w \in L^\infty(\mathbb{T})$. For simplicity, we often say that a function is outer (resp. inner) if it is equal to its outer (resp. inner) factor. We further set

$$H_m^2 := \{ \frac{g}{q_m} : g \in H^2, q_m \in \mathcal{P}_m, q_m(z) \neq 0 \text{ for } |z| \ge 1 \}.$$

Members of $H_m^2$ identify in $L^2$ with non-tangential limits of meromorphic functions with at most $m$ poles in $\mathbb{D}$ (counting multiplicities) whose $L^2$-means over $\{|z| = r\}$ remain eventually bounded as $r \to 1^-$. Two equivalent descriptions of $H_m^2$ are useful: on the one hand we have that $H_m^2 = B_m^{-1}H^2$, the set of quotients of $H^2$-functions by Blaschke products of degree at most $m$, on the other hand we get by pole-residue decomposition that $H_m^2 = H^2 + (\mathcal{R}_{m-1,m} \cap \bar{H}^{2,0})$. In a similar vein, we put

$$H_m^\infty := H_m^2 \cap L^\infty = B_m^{-1}H^\infty = \{ \frac{g}{q_m} : g \in H^\infty, q_m \in \mathcal{P}_m, q_m(z) \neq 0 \text{ for } |z| \ge 1 \}$$

for the set of meromorphic functions with at most $m$ poles in $L^\infty$.

# 4 Best rational and meromorphic approximation on the circle

We dwell on the discussion in [5]. For $n \ge 1$ an integer, the problem of best rational approximation of degree at most $n$ in $L^2$ is:

**Problem R(n):** Given $h \in L^2$, find $r^* \in \mathcal{R}_{n,n}$ such that

$$\|h - r^*\|_2 = \min_{r \in \mathcal{R}_{n,n}} \|h - r\|_2.$$

If we write in view of (10) the decomposition $h = h_1 + h_2$ with $h_1 \in H^2$, $h_2 \in \bar{H}^{2,0}$, and invoke partial fraction expansion to decompose an arbitrary $r \in \mathcal{R}_{n,n}$ as $r_1+r_2$ with $r_1 \in H^2$, $r_2 \in \bar{H}^{2,0}$, $\deg r_1+\deg r_2 \le n$, then by Parseval's relation

$$\|h - r\|_2^2 = \|h_1 - r_1\|_2^2 + \|h_2 - r_2\|_2^2$$

so that $R(n)$ reduces, modulo a combinatorial allocation of the degrees of $r_1$ and $r_2$ ($n+1$ choices), to a pair of problems of the following types:

**Problem RA(n):** Given $f \in H^2$, find $r^* \in \mathcal{R}_{n,n} \cap H^2$ such that

$$\|f - r^*\|_2 = \min_{r \in \mathcal{R}_{n,n} \cap H^2} \|f - r\|_2.$$
**Problem RAB(n):** Given $f \in \bar{H}^{2,0}$, find $r^* \in \mathcal{R}_{n-1,n} \cap \bar{H}^{2,0}$ such that

$$ \|f - r^*\|_2 = \min_{r \in \mathcal{R}_{n-1,n} \cap \bar{H}^{2,0}} \|f - r\|_2. $$

In “RA(n)” and “RAB(n)”, the letter “A” is mnemonic for “analytic” and “B” is mnemonic for “bar”.

Problem RA(n) is in fact equivalent to RAB(n). For we can parametrize $r \in \mathcal{R}_{n,n} \cap H^2$ as $r(0) + zr_3$ where $r(0) \in \mathbb{R}$ and $r_3 \in \mathcal{R}_{n-1,n} \cap H^2$ vary independently, hence by Parseval’s theorem

$$ \|f - r\|_2^2 = |f(0) - r(0)|^2 + \|(f - f(0)) - zr_3\|_2^2 $$

so that $r(0) = f(0)$ is the optimal choice. Then, since multiplying by $1/z$ preserves the modulus on $\mathbb{T}$, we find upon replacing $f$ by $(f - f(0))/z$ that Problem RA(n) is equivalent to the normalized version:

**Problem RAN(n):** Given $f \in H^2$, find $r^* \in \mathcal{R}_{n-1,n} \cap H^2$ such that

$$ \|f - r^*\|_2 = \min_{r \in \mathcal{R}_{n-1,n} \cap H^2} \|f - r\|_2. $$

Now, applying the check operation defined in (11) which preserves $\mathcal{R}_{n-1,n}$ and the degree, RAN(n) is mechanically equivalent to RAB(n), which proves the desired equivalence. Note that when passing from RA(n) to RAB(n), the initial $f \in H^2$, to be approximated from $\mathcal{R}_{n,n} \cap H^2$, gets transformed into the function $(f - f(0))^{\#} = f(1/\bar{z}) - f(0) \in \bar{H}^{2,0}$, to be approximated from $\mathcal{R}_{n-1,n} \cap \bar{H}^{2,0}$.

Finally, we also state the best meromorphic approximation problem with at most n poles in $L^2$:

**Problem MA(n):** Given $f \in L^2$, find $g^* \in H_n^2$ such that

$$ \|f - g^*\|_2 = \min_{g \in H_n^2} \|f - g\|_2. $$

Problem MA(n) is also equivalent to RAB(n). Indeed, since $H_n^2 = H^2 + (\mathcal{R}_{n-1,n} \cap \bar{H}^{2,0})$, it holds by orthogonality of $H^2$ and $\bar{H}^{2,0}$ that the $H^2$-component of a minimizer in MA(n) must be $\mathbf{P}_+(f)$ while the $\bar{H}^{2,0}$-component of this minimizer is a solution to RAB(n), with $f$ replaced by $\mathbf{P}_-(f)$.

It is known that RAB(n) has a solution which needs not be unique, and every solution has exact degree n unless f is rational of degree at most $n-1$ [8, 11, 3].

We shall write $d_2(f, \mathcal{R}_{n-1,n})$ (resp. $d_2(f, \mathcal{R}_{n,n})$) for the distance from $f$ to $\mathcal{R}_{n-1,n}$ (resp. $\mathcal{R}_{n,n}$) in $L^2$. Thus, if $f \in \bar{H}^{2,0}$, then $d_2(f, \mathcal{R}_{n-1,n})$ is both the value of Problem RAB(n) and of Problem MA(n); and if $f \in H^2$, then $d_2(f, \mathcal{R}_{n-1,n})$ (resp. $d_2(f, \mathcal{R}_{n,n})$) is the value of problem RAN(n) (resp. RA(n)). The value of MA(n) is denoted by $d_2(f, H_n^2)$.

Problems R(n) and MA(n) have natural analogs in $L^\infty$ provided that $f$ lies in that space: we denote by $d_\infty(f, \mathcal{R}_{n-1,n})$ (resp. $d_\infty(f, \mathcal{R}_{n,n})$) the distance from $f$ to $\mathcal{R}_{n-1,n}$ (resp. $\mathcal{R}_{n,n}$) in $L^\infty$; and by $d_\infty(f, H_n^\infty)$ the $L^\infty$-distance from $f$ to $H_n^\infty$. In $L^\infty$, however, no decomposition like (10) is valid and no splitting of the rational approximation problem into analogs of RA(n) and RAB(n) is available.

Of all these approximations problems, the $L^\infty$ analog of MA(n) may seem most difficult but still it is the only one known to reduce to matrix analysis, namely to singular value decomposition (at least when $f$ is rational). This is a consequence of the Adamjan-Arov-Krein theory (in short: AAK theory), a full account of which may be found in [1] or [12, Chapter 4]. For $f \in L^\infty$, these authors consider the Hankel operator $\Gamma_f$ with symbol $f$, which is the bounded operator defined as

$$ \Gamma_f: H^2 \longrightarrow \bar{H}^{2,0} \\ v \mapsto \mathbf{P}_-(fv), \qquad (17) $$

together with its singular values $s_0(\Gamma_f) \ge s_1(\Gamma_f) \ge s_2(\Gamma_f) \cdots$ given by

$$ s_n(\Gamma_f) := \inf\{|||\Gamma_f - L|||, L \in \mathcal{L}_n\}, \qquad (18) $$

where $\mathcal{L}_n$ is the collection of linear operators $H^2 \to \bar{H}^{2,0}$ with rank not exceeding n and $||| \cdot |||$ denotes the operator norm. Now, the AAK-theorem asserts that

$$ d_\infty(f, H_n^\infty) = s_n(\Gamma_f). \qquad (19) $$
Moreover, if $f \in C(\mathbb{T})$ then $\Gamma_f$ is compact, and the unique best approximant $g_n$ to $f$ from $H_n^\infty$ in $L^\infty$ is

$$g_n = \frac{\mathbf{P}_+(\Gamma_f(v_n))}{v_n} \qquad (20)$$

where $v_n$ is any singular vector associated with $s_n(\Gamma_f)$ (any eigenvector of $\Gamma_f^*\Gamma_f$ with eigenvalue $(s_n(\Gamma_f))^2$). When $\Gamma_f$ is compact, recall also that the Courant max min principle [13, sec. 22.11a] applies:

$$s_n(\Gamma_f) = \max_{V \in \mathcal{V}_{n+1}} \min_{\substack{v \in V \\ \|v\|_2 = 1}} \|\Gamma_f(v)\|_2, \qquad (21)$$

where $\mathcal{V}_{n+1}$ is the collection of linear subspaces of $H^2$ of complex dimension at least $n+1$.

Let us turn to problem RAB(n), which is equivalent to MA(n) as pointed out already. That is, let us move from meromorphic approximation in $L^\infty$ to meromorphic approximation in $L^2$. In this context, for $f \in \bar{H}^{2,0}$, we define again a Hankel operator $A_f$ with symbol $f$ by

$$A_f: H^\infty \to \bar{H}^{2,0} \\ v \mapsto \mathbf{P}_-(fv). \qquad (22)$$

Although the definitions of $A_f$ and $\Gamma_f$ are formally the same, observe that the domains in (22) and (17) are different. Evidently, $A_f$ is continuous and $|||A_f||| = \|f\|_2$. In fact $A_f$ is compact, because it is a limit of finite rank operators: indeed, since rational functions are dense in $\bar{H}^{2,0}$, we may find $r_n \in \mathcal{R}_{n-1,n}$ arbitrary close to $f$, and use Kronecker's theorem that Hankel operaors with rational symbol have finite rank [12].

The Theorem below is a special case of [4, thm. 8.1], but granted its importance in the present paper and its short proof, we nevertheless provide an argument for the ease of the reader.

**Theorem 1.** For $f \in \bar{H}^{2,0}$, it holds that

$$d_2(f, \mathcal{R}_{n-1,n}) = \min_{b_n \in B_n} \|A_f(b_n)\|_2. \qquad (23)$$

Moreover, a rational function $r_n \in \mathcal{R}_{n-1,n}$ is a solution to RAB(n) if, and only if

$$r_n = \frac{\mathbf{P}_+(fb_n)}{b_n} \qquad (24)$$

where $b_n$ is any minimizing Blaschke product in the right hand side of (23).

*Proof.* Let us parametrize $r \in \mathcal{R}_{n-1,n} \cap \bar{H}^{2,0}$ as $r = p_{n-1}/q_n$ where $p_{n-1}$ ranges over $\mathcal{P}_{n-1}$ and $q_n$ ranges over those polynomials in $\mathcal{P}_n$ whose roots all lie in $\mathbb{D}$. Then $q_n/\tilde{q}_n \in B_n$, so multiplication by $q_n/\tilde{q}_n$ is an isometry in $L^2$, and since $p_{n-1}/\tilde{q}_n \in H^2$ we get by orthogonality of $H^2$ and $\bar{H}^{2,0}$ on using that $\mathbf{P}_+ + \mathbf{P}_- = I$:

$$\|f - \frac{p_{n-1}}{q_n}\|_2^2 = \|f \frac{q_n}{\tilde{q}_n} - \frac{p_{n-1}}{\tilde{q}_n}\|_2^2 = \|\mathbf{P}_-(f \frac{q_n}{\tilde{q}_n})\|_2^2 + \|\mathbf{P}_+(f \frac{q_n}{\tilde{q}_n}) - \frac{p_{n-1}}{\tilde{q}_n}\|_2^2. \qquad (25)$$

Clearly the product of a $\bar{H}^{2,0}$-function by a polynomial in $\mathcal{P}_n$ yields a member of $z^n \bar{H}^{2,0}$. Therefore

$$\tilde{q}_n \mathbf{P}_+(f \frac{q_n}{\tilde{q}_n}) = f q_n - \tilde{q}_n \mathbf{P}_-(f \frac{q_n}{\tilde{q}_n}) \in z^n \bar{H}^{2,0} \cap H^2 = \mathcal{P}_{n-1}, \qquad (26)$$

implying that $p_{n-1} = \tilde{q}_n \mathbf{P}_+(f q_n / \tilde{q}_n)$ is the minimizing choice in (25) for fixed $q_n$. Consequently

$$\min_{r \in \mathcal{R}_{n-1,n} \cap \bar{H}^{2,0}} \|f - r\|_2 = \min_{q_n \in \mathcal{P}_n, \mathcal{Z}(q_n) \subset \mathbb{D}} \|\mathbf{P}_-(f \frac{q_n}{\tilde{q}_n})\|_2 = \min_{b_n \in B_n} \|\mathbf{P}_-(fb_n)\|_2 = \min_{b_n \in B_n} \|A_f(b_n)\|_2. \qquad (27)$$

That the infimum is indeed attained in (27) follows from (25) and the fact that RAB(n) has a solution. $\square$
We digress at this point to stress how Theorem 1 parallels AAK theory: we may define the singular values $s_n(A_f)$ by (18), only with $A_f$ instead of $\Gamma_f$ and $H^\infty$ instead of $H^{21}$, and still $d_2(f, \mathcal{R}_{n-1,n}) = s_n(A_f)$; i.e. the natural analog of (19) holds. However, a direct analog of the Courant max-min principle (21) does not hold when we replace $\Gamma_f$ by $A_f$ and $H^2$ by $H^\infty$ in the definition of $\mathcal{V}_{n+1}$. But still, a nonlinear analog of the max-min principle does hold if the collection of all Euclidean unit spheres of real dimension $2n+1$ in $H^2$, over which the maximizing step is made in (21), gets replaced by the collection of all weak-* closed symmetric subsets of the unit sphere of $H^\infty$ whose genus is at least $2n+2^2$. Specifically, if we let $\mathcal{K}_{2n+2}^\infty$ denote the collection of all such sets, it holds that

$$s_n(A_f) = \max_{K \in \mathcal{K}_{2n+2}^\infty} \min_{u \in K} \|A_f(u)\|_2, \quad n = 0, 1, 2, \dots, \qquad (28)$$

and (23) indicates that $B_n$ is an optimal weak-* closed subset of the unit sphere of $H^\infty$ with genus no less than (in fact equal to) $2n+2$ in (28). Moreover, if $b_n$ is a minimizing Blaschke product in (28), formula (24) stands analog to (20). We refer the reader to [4] for an account of best meromorphic approximation in $L^p$ for $2 \le p \le \infty$ along these lines.

# 5 $H^2$ rational approximation to $z^N$: lower bounds

When $h$ is continuous of constant modulus on $\mathbb{T}$, the unique best approximant to $h$ from $\mathcal{R}_{m,n}$ in $L^\infty$ must be zero unless $-n \le W(h) \le m$. Indeed, if $r_{m,n} \in \mathcal{R}_{m,n}$ is such that $|h - r_{m,n}| < |h|$ on $\mathbb{T}$, then $[0, 1] \ni t \mapsto h + t(r_{m,n} - h)$ is a nevervanishing homotopy between $h$ and $r_{m,n}$, therefore these must have the same index and yet $-n \le W(r_{m,n}) \le m$, by (5). In particular, for $f \in B_N$ with $N$ finite, the solution to problem RA($n$) is zero when $N > n$; that is to say, $d_\infty(f, \mathcal{R}_{n,n}) = 1$ if $n < N < \infty$; the same is in fact true when $N = \infty$, though in this case $f$ is no longer continuous and the argument needs adjustment, see [5, Lemma 1].

The situation is different if we consider best rational approximation in $L^2$. Indeed, if $f \in B_N$ then $d_2(f, \mathcal{R}_{n,n}) < 1$, because any solution to RAB($n$) has exact degree $n$ unless $N < n$, as mentioned in Section 4; hence, zero is never a best approximant. Of course, since approximation in $L^\infty$ is ineffective, one may surmise that $d_2(f, \mathcal{R}_{n,n})$ cannot be small when $f \in B_N$ and $n$ is much smaller than $N$, but this suspicion calls for quantification. Now, if $(\zeta_j)_{1 \le j \le N}$ is the sequence of zeros of $f \in B_N$ with moduli arranged in nondecreasing order ($N$ may be infinite), it can be shown building on Theorem 1 and AAK theory that $d_2(f, \mathcal{R}_{n,n})$ is no less than $\sqrt{(1 - |\zeta_{n+1}|^2)/(n+1)}$ provided that $n < N$; see [5, Theorem 5]. Since $(1 - |\zeta_{n+1}|)$ can be the general term of a convergent series with positive, non-increasing but otherwise arbitrary summands, this estimate gives one an appraisal of how slow the decay of $d_2(f, \mathcal{R}_{n,n})$ to zero can be as $n \to \infty$ when $N$ is infinite. Still, it is unclear how sharp this lower bound, and when $N$ is finite it gives no information on the speed of $L^2$-approximation as $n$ increases to $N$ (when $n > N$, the best approximant is of course $f$ itself). This is the question that we take up now. Below we consider the particular case where $f = z^N$, which is of special significance from a system-theoretic viewpoint, because it quantifies how poorly discrete-time delay systems lend themselves to model reduction.

**Theorem 2.** Let $0 \le n < N$ be integers. Then,

$$d_2(z^N, \mathcal{R}_{n,n}) \ge \sqrt{1 - \frac{n}{N}}. \qquad (29)$$

*Proof.* It follows from Theorem 1 and the discussion of Problem RAN($n$) in Section 4 that

$$d_2(z^N, \mathcal{R}_{n,n}) = d_2(z^{-N}, \mathcal{R}_{n-1,n}) = \min_{b_n \in B_n} \|A_{z^{-N}}(b_n)\|_2 = \min_{b_n \in B_n} \|\mathbf{P}_-(z^{-N}b_n)\|_2. \qquad (30)$$

<footnote n="1">¹In [4, Section 8] it is required in this case that members of $\mathcal{L}_n$ should be weak-* continuous $H^\infty \to \tilde{H}^{2,0}$, however this is redundant because clearly we may restrict to continuous operators, and those of finite rank are automatically weak-* continuous.</footnote>

<footnote n="2">²The genus of a symmetric set $K$ (meaning that $-v \in K$ when $v \in K$) is the smallest positive integer for which there is an odd continuous mapping $K \to \mathbb{R}^n \setminus \{0\}$ [13]; by the Borsuk-Ulam theorem, Euclidean spheres of real dimension $2n+1$ have genus $2n+2$.</footnote>
Pick $b_n \in B_n$ and write its Fourier expansion as $b_n = \sum_{k=0}^{+\infty} a_k z^k$. Then, (12) and Parseval's identity yield

$$ \|\mathbf{P}_-(z^{-N}b_n)\|_2^2 = \sum_{k=0}^{N-1} |a_k|^2. \qquad (31) $$

In another connection, since $e^{-iN\theta}b_n(e^{i\theta}) = \sum_{k=0}^{+\infty} a_k e^{i(k-N)\theta}$, we get from (5) and (7) that

$$ W(z^{-N}b_n) = n - N = \sum_{k=0}^{N-1} (k-N)|a_k|^2 + \sum_{k=N}^{+\infty} (k-N)|a_k|^2; \qquad (32) $$

note that (32) is valid even if $b_n$ has a zero at zero, because the quantity $n - N$ remains unchanged if $z^N$ and $b_n$ undergo the same drop in degree. From (32) we deduce at once that

$$ N - n \le \sum_{k=0}^{N-1} (N - k) |a_k|^2, $$

whence

$$ 1 - \frac{n}{N} \le \sum_{k=0}^{N-1} \left(1 - \frac{k}{N}\right) |a_k|^2 \le \sum_{k=0}^{N-1} |a_k|^2. \qquad (33) $$

Inequality (29) now follows from (33), (31) and (30). $\square$

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