# Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design

Authors: Christophe Chesneau; Maher Kachour
Journal: Advances in Statistics
 (2015)

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## Abstract

A convolution regression model with random design is considered. We investigate the estimation of the derivatives of an unknown function, element of the convolution product. We introduce new estimators based on wavelet methods and provide theoretical guarantees on their good performances.


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## Body

## 1. Introduction

We consider the convolution regression model with random design described as follows. Let(Y1,X1),…,(Yn,Xn) be n i.i.d. random variables defined on a probability space (Ω,A,P), where(1)Yv=(f⋆g)(Xv)+ξv,v=1,…,n,(f⋆g)(x)=∫-∞∞‍f(t)g(x-t)dt, f:R→R is an unknown function, g:R→R is a known function, X1,…,Xn are n i.i.d. random variables with common density h:R→[0,∞), and ξ1,…,ξn are n i.i.d. random variables such that E(ξ1)=0 and E(ξ12)<∞. Throughout this paper, we assume that f, g, and h are compactly supported with supp(f)=[a,b], supp(g)=[a′,b′], supp(h)=[a*,b*], (a,b,a′,b′)∈R4, a<b, a′<b′, a*=a+a′, b*=b+b′, f is m times differentiable with m∈N, g is integrable and ordinary smooth (the precise definition is given by(K2) in Section 3.1), and Xv and ξv are independent for any v=1,…,n. We aim to estimate the unknown function f and its mth derivative, denoted by f(m), from the sample (Y1,X1),…,(Yn,Xn).The motivation of this problem is the deconvolution of a signalf from f⋆g perturbed by noise and randomly observed. The function g can represent a driving force that was applied to a physical system. Such situations naturally appear in various applied areas, as astronomy, optics, seismology, and biology. Model (1) can also be viewed as a natural extension of some 1-periodic convolution regression models as those considered by, for example, Cavalier and Tsybakov [1], Pensky and Sapatinas [2], and Loubes and Marteau [3]. In the form (1), it has been considered in Bissantz and Birke [4] and Birke et al. [5] with a deterministic design and in Hildebrandt et al. [6] with a random design. These last works focus on kernel methods and establish their asymptotic normality. The estimation of f(m), more general to f=f(0), is of interest to examine possible bumps and to study the convexity-concavity properties of f (see, for instance, Prakasa Rao [7], for standard statistical models).In this paper, we introduce new estimators forf(m) based on wavelet methods. Through the use of a multiresolution analysis, these methods enjoy local adaptivity against discontinuities and provide efficient estimators for a wide variety of unknown functions f(m). Basics on wavelet estimation can be found in, for example, Antoniadis [8], Härdle et al. [9], and Vidakovic [10]. Results on the wavelet estimation of f(m) in other regression frameworks can be found in, for example, Cai [11], Petsa and Sapatinas [12], and Chesneau [13].The first part of the study is devoted to the case whereh, the common density of X1,…,Xn, is known. We develop a linear wavelet estimator and an adaptive nonlinear wavelet estimator. The second one uses the double hard thresholding technique introduced by Delyon and Juditsky [14]. It does not depend on the smoothness of f(m) in its construction; it is adaptive. We exhibit their rates of convergence via the mean integrated squared error (MISE) and the assumption that f(m) belongs to Besov balls. The obtained rates of convergence coincide with existing results for the estimation of f(m) in the 1-periodic convolution regression models (see, for instance, Chesneau [15]).The second part is devoted to the case whereh is unknown. We construct a new linear wavelet estimator using a plug-in approach for the estimation of h. Its construction follows the idea of the “NES linear wavelet estimator” introduced by Pensky and Vidakovic [16] in another regression context. Then we investigate its MISE properties when f(m) belongs to Besov balls, which naturally depend on the MISE of the considered estimator for h. Furthermore, let us mention that all our results are proved with only moments of order 2 on ξ1, which provides another theoretical contribution to the subject.The remaining part of this paper is organized as follows. In Section2 we describe some basics on wavelets and Besov balls and present our wavelet estimation methodology. Section 3 is devoted to our estimators and their performances. The proofs are carried out in Section 4.


## 2. Preliminaries

This section is devoted to the presentation of the considered wavelet basis, the Besov balls, and our wavelet estimation methodology.


### 2.1. Wavelet Basis

Let us briefly present the wavelet basis on the interval[a,b], (a,b)∈R2, introduced by Cohen et al. [17]. Let ϕ and ψ be the initial wavelet functions of the Daubechies wavelets familydb2N with N≥1 (see, e.g., Daubechies [18]). These functions have the distinction of being compactly supported and belong to the class Ca for N>5a. For any j≥0 and k∈Z, we set (2)ϕj,k(x)=2j/2ϕ(2jx-k),ψj,k(x)=2j/2ψ(2jx-k).With appropriated treatments at the boundaries, there exist an integerτ and a set of consecutive integers Λj of cardinality proportional to 2j (both depending on a, b, and N) such that, for any integer l≥τ, (3)B={ϕl,k,k∈Λl;ψj,k;j∈N-{0,…,l-1},k∈Λj}forms an orthonormal basis of the space of squared integrable functions on [a,b]; that is, (4)L2([a,b])=u:[a,b]⟶R;∫ab‍ux2dx1/2<∞.For the case a=0 and b=1, τ is the smallest integer satisfying 2τ≥2N and Λj={0,…,2j-1}.For any integerl≥τ and u∈L2([a,b]), we have the following wavelet expansion: (5)ux=∑k∈Λl‍cl,kϕl,kx+∑j=l∞∑k∈Λj‍dj,kψj,kx,x∈a,b,where(6)cj,k=∫ab‍u(x)ϕj,k(x)dx,dj,k=∫ab‍u(x)ψj,k(x)dx.An interesting feature of the wavelet basis is to provide sparse representation of u; only few wavelet coefficients dj,k characterized by a high magnitude reveal the main details of u. See, for example, Cohen et al. [17] and Mallat [19].


### 2.2. Besov Balls

We say that a functionu∈L2([a,b]) belongs to the Besov ball Bp,rs(M) with s>0, p≥1, r≥1, and M>0 if there exists a constant C>0 such that cj,k and dj,k (6) satisfy (7)2τ1/2-1/p∑k∈Λτ‍|cτ,k|p1/p+∑j=τ∞‍2j(s+1/2-1/p)∑k∈Λj‍dj,kp1/pr1/r≤C,with the usual modifications if p=∞ or r=∞.The interest of Besov balls is to contain various kinds of homogeneous and inhomogeneous functionsu. See, for example, Meyer [20], Donoho et al. [21], and Härdle et al. [9].


### 2.3. Wavelet Estimation

Letf be the unknown function in (1) and B the considered wavelet basis taken with N>5m (to ensure that ϕ and ψ belong to the class Cm). Suppose that f(m) exists with f(m)∈L2([a,b]).The first step in the wavelet estimation consists in expandingf(m) on B as(8)fmx=∑k∈Λl‍cl,kmϕl,kx+∑j=l∞‍∑k∈Λj‍dj,kmψj,kx,x∈a,b,where l≥τ and(9)cj,k(m)=∫ab‍f(m)(x)ϕj,k(x)dx,dj,k(m)=∫ab‍f(m)(x)ψj,k(x)dx.The second step is the estimation ofcj,k(m) and dj,k(m) using (Y1,X1),…,(Yn,Xn). The idea of the third step is to exploit the sparse representation of f(m) by selecting the most interesting wavelet coefficients estimators. This selection can be of different natures (truncation, thresholding,…). Finally, we reconstruct these wavelet coefficients estimators on B, providing an estimator f^(m) for f(m).In this study, we evaluate the performance off^(m) by studying the asymptotic properties of its MISE under the assumption that f(m)∈Bp,rs(M). More precisely, we aim to determine the sharpest rate of convergence ωn such that (10)E∫ab‍f^mx-fmx2dx≤Cωn,where C denotes a constant independent of n.


## 2.1. Wavelet Basis

Let us briefly present the wavelet basis on the interval[a,b], (a,b)∈R2, introduced by Cohen et al. [17]. Let ϕ and ψ be the initial wavelet functions of the Daubechies wavelets familydb2N with N≥1 (see, e.g., Daubechies [18]). These functions have the distinction of being compactly supported and belong to the class Ca for N>5a. For any j≥0 and k∈Z, we set (2)ϕj,k(x)=2j/2ϕ(2jx-k),ψj,k(x)=2j/2ψ(2jx-k).With appropriated treatments at the boundaries, there exist an integerτ and a set of consecutive integers Λj of cardinality proportional to 2j (both depending on a, b, and N) such that, for any integer l≥τ, (3)B={ϕl,k,k∈Λl;ψj,k;j∈N-{0,…,l-1},k∈Λj}forms an orthonormal basis of the space of squared integrable functions on [a,b]; that is, (4)L2([a,b])=u:[a,b]⟶R;∫ab‍ux2dx1/2<∞.For the case a=0 and b=1, τ is the smallest integer satisfying 2τ≥2N and Λj={0,…,2j-1}.For any integerl≥τ and u∈L2([a,b]), we have the following wavelet expansion: (5)ux=∑k∈Λl‍cl,kϕl,kx+∑j=l∞∑k∈Λj‍dj,kψj,kx,x∈a,b,where(6)cj,k=∫ab‍u(x)ϕj,k(x)dx,dj,k=∫ab‍u(x)ψj,k(x)dx.An interesting feature of the wavelet basis is to provide sparse representation of u; only few wavelet coefficients dj,k characterized by a high magnitude reveal the main details of u. See, for example, Cohen et al. [17] and Mallat [19].


## 2.2. Besov Balls

We say that a functionu∈L2([a,b]) belongs to the Besov ball Bp,rs(M) with s>0, p≥1, r≥1, and M>0 if there exists a constant C>0 such that cj,k and dj,k (6) satisfy (7)2τ1/2-1/p∑k∈Λτ‍|cτ,k|p1/p+∑j=τ∞‍2j(s+1/2-1/p)∑k∈Λj‍dj,kp1/pr1/r≤C,with the usual modifications if p=∞ or r=∞.The interest of Besov balls is to contain various kinds of homogeneous and inhomogeneous functionsu. See, for example, Meyer [20], Donoho et al. [21], and Härdle et al. [9].


## 2.3. Wavelet Estimation

Letf be the unknown function in (1) and B the considered wavelet basis taken with N>5m (to ensure that ϕ and ψ belong to the class Cm). Suppose that f(m) exists with f(m)∈L2([a,b]).The first step in the wavelet estimation consists in expandingf(m) on B as(8)fmx=∑k∈Λl‍cl,kmϕl,kx+∑j=l∞‍∑k∈Λj‍dj,kmψj,kx,x∈a,b,where l≥τ and(9)cj,k(m)=∫ab‍f(m)(x)ϕj,k(x)dx,dj,k(m)=∫ab‍f(m)(x)ψj,k(x)dx.The second step is the estimation ofcj,k(m) and dj,k(m) using (Y1,X1),…,(Yn,Xn). The idea of the third step is to exploit the sparse representation of f(m) by selecting the most interesting wavelet coefficients estimators. This selection can be of different natures (truncation, thresholding,…). Finally, we reconstruct these wavelet coefficients estimators on B, providing an estimator f^(m) for f(m).In this study, we evaluate the performance off^(m) by studying the asymptotic properties of its MISE under the assumption that f(m)∈Bp,rs(M). More precisely, we aim to determine the sharpest rate of convergence ωn such that (10)E∫ab‍f^mx-fmx2dx≤Cωn,where C denotes a constant independent of n.


## 3. Rates of Convergence

In this section, we list the assumptions on the model, present our wavelet estimators, and determine their rates of convergence under the MISE over Besov balls.


### 3.1. Assumptions

Let us recall thatf and g are the functions in (1) and h is the density of X1.We formulate the following assumptions.(K1)
We havef(q)(a)=f(q)(b)=0 for any q∈{0,…,m}, f(m)∈L2([a,b]), and there exists a known constant C1>0 such that sup⁡x∈[a,b]|f(x)|≤C1.(K2)
First of all, let us define the Fourier transform of an integrable functionu by (11)F(u)(x)=∫-∞∞‍u(y)e-ixydy,x∈R.The notation·¯ will be used for the complex conjugate.We haveg∈L2([a′,b′]) and there exist two constants, c1>0 and δ≥0, such that(12)Fgx≥c11+x2δ/2,x∈R.(K3)
There exists a constantc2>0 such that (13)c2≤inf⁡x∈[a*,b*]h(x).The assumptions(K1) and(K3) are standard in a nonparametric regression framework (see, for instance, Tsybakov [22]). Remark that we do not need f(a)=f(b)=0 for the estimation of f=f(0). The assumption(K2) is the so-called “ordinary smooth case” on g. It is common for the deconvolution-estimation of densities (see, e.g., Fan and Koo [23] and Pensky and Vidakovic [24]). An example of compactly supported function g satisfying(K2) is g(x)=∫12‍ymax⁡1-|x|/y,0dy. Then supp(g)=[-2,2], g∈L2([-2,2]), and(K2) is satisfied with δ=2 and c1=min⁡(inf⁡x∈[-2π,2π]|F(g)(x)|,1/(4π2))>0.


### 3.2. Whenh Is Known

#### 3.2.1. Linear Wavelet Estimator

We define the linear wavelet estimatorf^1(m) by(14)f^1(m)(x)=∑k∈Λj0‍c^j0,k(m)ϕj0,k(x),x∈[a,b],where(15)c^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixXvdx,and j0 is an integer chosen a posteriori.Proposition1 presents an elementary property of c^j,k(m).Proposition 1.
Letc^j,k(m) be (15) and let cj,k(m) be (9). Suppose that(K1) holds. Then one has (16)E(c^j,k(m))=cj,k(m).Theorem2 below investigates the performance of f^1(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 2.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, p≥1, r≥1, s∈(max⁡(1/p-1/2,0),N), and N>5(m+δ+1). Let f^1(m) be defined by (14) with j0 such that(17)2j0=[n1/(2s*+2m+2δ+1)],
s
*
=
s
+
min
⁡
(
1
/
2
-
1
/
p
,
0
) ([a] denotes the integer part of a).
Then there exists a constantC>0 such that (18)E∫ab‍f^1mx-fmx2dx≤Cn-2s*/(2s*+2m+2δ+1).Note that the rate of convergencen-2s*/(2s*+2m+2δ+1) corresponds to the one obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted linear wavelet estimator (see, e.g., Chesneau [15]).The considered estimatorf^1(m) depends on s (the smoothness parameter of f(m)); it is not adaptive. This aspect, as well as the rate of convergence n-2s*/(2s*+2m+2δ+1), can be improved with thresholding methods. The next paragraph is devoted to one of them: the hard thresholding method.


#### 3.2.2. Hard Thresholding Wavelet Estimator

Suppose that(K2) is satisfied. We define the hard thresholding wavelet estimator f^2(m) by(19)f^2mx=∑k∈Λτ‍c^τ,kmϕτ,kx+∑j=τj1‍∑k∈Λj‍d^j,k(m)1|d^j,k(m)|≥κλjψj,k(x),x∈[a,b], where c^j,k(m) is defined by (15),(20)d^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFψj,k¯(x)F(g)(x)e-ixXvdx·1(Yv/hXv)(1/2π)∫-∞∞‍(ix)m((Fψj,k¯(x))/(F(g)(x)))e-ixXvdx≤ςj,1 is the indicator function, κ>0 is a large enough constant, j1 is the integer satisfying (21)2j1=n1/(2m+2δ+1),δ refers to (12), (22)ςj=θψ2mj2δjnln⁡n,λj=θψ2mj2δjln⁡nn,θψ=1πc2c12C12∫-∞∞‍gxdx2+E(ξ12)C12∫-∞∞‍gxdx2+E(ξ12)·∫-∞∞‍xm1+x2δFψx2dx1/2.The construction off^2(m) uses the double hard thresholding technique introduced by Delyon and Juditsky [14] and recently improved by Chaubey et al. [25]. The main interest of the thresholding using λj is to make f^2(m) adaptive; the construction (and performance) of f^2(m) does not depend on the knowledge of the smoothness of f(m). The role of the thresholding using ςj in (20) is to relax some usual restrictions on the model. To be more specific, it enables us to only suppose that ξ1 admits finite moments of order 2 (with known E(ξ12) or a known upper bound of E(ξ12)), relaxing the standard assumption E(|ξ1|k)<∞, for any k∈N.Further details on the constructions of hard thresholding wavelet estimators can be found in, for example, Donoho and Johnstone [26, 27], Donoho et al. [21, 28], Delyon and Juditsky [14], and Härdle et al. [9].Theorem3 below investigates the performance of f^2(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 3.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, r≥1, {p≥2, s∈(0,N)} or {p∈[1,2),s∈((2m+2δ+1)/p,N)}, and N>5(m+δ+1). Let f^2(m) be defined by (19). Then there exists a constant C>0 such that (23)E∫ab‍f^2mx-fmx2dx≤Cln⁡nn2s/(2s+2m+2δ+1).The proof of Theorem3 is an application of a general result established by [25, Theorem 6.1]. Let us mention that ln⁡n/n2s/(2s+2m+2δ+1) corresponds to the rate of convergence obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau [15]). In the case m=0 and δ=0, this rate of convergence becomes the optimal one in the minimax sense for the standard density-regression estimation problems (see Härdle et al. [9]).In comparison to Theorem2, note that(i)
for the casep≥2 corresponding to the homogeneous zone of Besov balls (ln⁡n/n)2s/(2s+2m+2δ+1) is equal to the rate of convergence attained by f^1(m) up to a logarithmic term,(ii)
for the casep∈[1,2) corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.


### 3.3. Whenh Is Unknown

In the case whereh is unknown, we propose a plug-in technique which consists in estimating h in the construction of f^1(m)  (14). This yields the linear wavelet estimator f^3(m) defined by(24)f^3(m)(x)=∑k∈Λj2‍c~j2,k(m)ϕj2,k(x),x∈[a,b],where(25)c~j,km=1an∑v=1an‍Yvh^Xv1h^Xv≥c2/212π·∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixXvdx,an=[n/2], j2 is an integer chosen a posteriori, c2 refers to(K3), and h^ is an estimator of h constructed from the random variables Un=(Xan+1,…,Xn).There are numerous possibilities for the choice ofh^. For instance, h^ can be a kernel density estimator or a wavelet density estimator (see, e.g., Donoho et al. [21], Härdle et al. [9], and Juditsky and Lambert-Lacroix [29]).The estimatorf^3(m) is derived to the “NES linear wavelet estimator” introduced by Pensky and Vidakovic [16] and recently revisited in a more simple form by Chesneau [13].Theorem4 below determines an upper bound of the MISE of f^3(m).Theorem 4.
Suppose that(K1)–(K3) are satisfied, h∈L2([a*,b*]), and that f(m)∈Bp,rs(M) with M>0, p≥1, r≥1, s∈(max⁡(1/p-1/2,0),N), and N>5(m+δ+1). Let f^3(m) be defined by (24) with j2 such that 2j2≤n. Then there exists a constant C>0 such that (26)E∫ab‍f^3mx-fmx2dx≤C2(2m+2δ+1)j2max⁡E∫a*b*‍h^x-hx2dx,1n∫a*b*‍h^x-hx2dx+2-2j2s*,with s*=s+min⁡(1/2-1/p,0).The proof follows the idea of [13, Theorem 3] and uses technical operations on Fourier transforms.From Theorem4,(i)
if we choseh^=h and j2=j0 (17), we obtain Theorem 2,(ii)
ifh^ and h satisfy that there exist υ∈[0,1] and a constant C>0 such that (27)E∫a*b*‍h^x-hx2dx≤Cn-υ,then, the optimal integer j2 is such that 2j2=[nυ/(2s*+2m+2δ+1)] and we obtain the following rate of convergence for f^3(m): (28)E∫ab‍f^3mx-fmx2dx≤Cn-2s*υ/(2s*+2m+2δ+1).Naturally the estimation of h has a negative impact on the performance of f^3(m). In particular, if h∈Bp′,r′s′(M′), then the standard density linear wavelet estimator h^ attains the rate of convergence n-υ with υ=2s∘/(2s∘+1) and s∘=s′+min⁡(1/2-1/p′,0) (and it is optimal in the minimax sense for p′≥2; see Härdle et al. [9]). With this choice, the rate of convergence for f^3(m) becomes n-4s*s∘/(2s∘+1)(2s*+2m+2δ+1). Let us mention that f^3(m) is not adaptive since it depends on s. However, f^3(m) remains an acceptable first approach for the estimation of f(m) with unknown h.Conclusion and Perspectives. This study considers the estimation of f(m) from (1). According to the knowledge of h or not, we propose wavelet methods and prove that they attain fast rates of convergence under the MISE over Besov balls. Among the perspectives of this work, we retain the following.(i)
The relaxation of the assumption(K2), perhaps by considering(K2′): there exist four constants, C1>0, ω∈N, η>0, and δ≥0, such that (29)Fgx-1≤C1sinπxη-ω1+xδ,x∈R.This condition was first introduced by Delaigle and Meister [30] in a context of deconvolution-estimation of function. It implies(K2) and has the advantage to consider some functions g having zeros in Fourier transform domain as numerous kinds of compactly supported functions.(ii)
The construction of an adaptive version off^3(m) through the use of a thresholding method.(iii)
The extension of our results to theLp risk with p≥1.All these aspects need further investigations that we leave for future works.


## 3.1. Assumptions

Let us recall thatf and g are the functions in (1) and h is the density of X1.We formulate the following assumptions.(K1)
We havef(q)(a)=f(q)(b)=0 for any q∈{0,…,m}, f(m)∈L2([a,b]), and there exists a known constant C1>0 such that sup⁡x∈[a,b]|f(x)|≤C1.(K2)
First of all, let us define the Fourier transform of an integrable functionu by (11)F(u)(x)=∫-∞∞‍u(y)e-ixydy,x∈R.The notation·¯ will be used for the complex conjugate.We haveg∈L2([a′,b′]) and there exist two constants, c1>0 and δ≥0, such that(12)Fgx≥c11+x2δ/2,x∈R.(K3)
There exists a constantc2>0 such that (13)c2≤inf⁡x∈[a*,b*]h(x).The assumptions(K1) and(K3) are standard in a nonparametric regression framework (see, for instance, Tsybakov [22]). Remark that we do not need f(a)=f(b)=0 for the estimation of f=f(0). The assumption(K2) is the so-called “ordinary smooth case” on g. It is common for the deconvolution-estimation of densities (see, e.g., Fan and Koo [23] and Pensky and Vidakovic [24]). An example of compactly supported function g satisfying(K2) is g(x)=∫12‍ymax⁡1-|x|/y,0dy. Then supp(g)=[-2,2], g∈L2([-2,2]), and(K2) is satisfied with δ=2 and c1=min⁡(inf⁡x∈[-2π,2π]|F(g)(x)|,1/(4π2))>0.


## 3.2. Whenh Is Known

### 3.2.1. Linear Wavelet Estimator

We define the linear wavelet estimatorf^1(m) by(14)f^1(m)(x)=∑k∈Λj0‍c^j0,k(m)ϕj0,k(x),x∈[a,b],where(15)c^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixXvdx,and j0 is an integer chosen a posteriori.Proposition1 presents an elementary property of c^j,k(m).Proposition 1.
Letc^j,k(m) be (15) and let cj,k(m) be (9). Suppose that(K1) holds. Then one has (16)E(c^j,k(m))=cj,k(m).Theorem2 below investigates the performance of f^1(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 2.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, p≥1, r≥1, s∈(max⁡(1/p-1/2,0),N), and N>5(m+δ+1). Let f^1(m) be defined by (14) with j0 such that(17)2j0=[n1/(2s*+2m+2δ+1)],
s
*
=
s
+
min
⁡
(
1
/
2
-
1
/
p
,
0
) ([a] denotes the integer part of a).
Then there exists a constantC>0 such that (18)E∫ab‍f^1mx-fmx2dx≤Cn-2s*/(2s*+2m+2δ+1).Note that the rate of convergencen-2s*/(2s*+2m+2δ+1) corresponds to the one obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted linear wavelet estimator (see, e.g., Chesneau [15]).The considered estimatorf^1(m) depends on s (the smoothness parameter of f(m)); it is not adaptive. This aspect, as well as the rate of convergence n-2s*/(2s*+2m+2δ+1), can be improved with thresholding methods. The next paragraph is devoted to one of them: the hard thresholding method.


### 3.2.2. Hard Thresholding Wavelet Estimator

Suppose that(K2) is satisfied. We define the hard thresholding wavelet estimator f^2(m) by(19)f^2mx=∑k∈Λτ‍c^τ,kmϕτ,kx+∑j=τj1‍∑k∈Λj‍d^j,k(m)1|d^j,k(m)|≥κλjψj,k(x),x∈[a,b], where c^j,k(m) is defined by (15),(20)d^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFψj,k¯(x)F(g)(x)e-ixXvdx·1(Yv/hXv)(1/2π)∫-∞∞‍(ix)m((Fψj,k¯(x))/(F(g)(x)))e-ixXvdx≤ςj,1 is the indicator function, κ>0 is a large enough constant, j1 is the integer satisfying (21)2j1=n1/(2m+2δ+1),δ refers to (12), (22)ςj=θψ2mj2δjnln⁡n,λj=θψ2mj2δjln⁡nn,θψ=1πc2c12C12∫-∞∞‍gxdx2+E(ξ12)C12∫-∞∞‍gxdx2+E(ξ12)·∫-∞∞‍xm1+x2δFψx2dx1/2.The construction off^2(m) uses the double hard thresholding technique introduced by Delyon and Juditsky [14] and recently improved by Chaubey et al. [25]. The main interest of the thresholding using λj is to make f^2(m) adaptive; the construction (and performance) of f^2(m) does not depend on the knowledge of the smoothness of f(m). The role of the thresholding using ςj in (20) is to relax some usual restrictions on the model. To be more specific, it enables us to only suppose that ξ1 admits finite moments of order 2 (with known E(ξ12) or a known upper bound of E(ξ12)), relaxing the standard assumption E(|ξ1|k)<∞, for any k∈N.Further details on the constructions of hard thresholding wavelet estimators can be found in, for example, Donoho and Johnstone [26, 27], Donoho et al. [21, 28], Delyon and Juditsky [14], and Härdle et al. [9].Theorem3 below investigates the performance of f^2(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 3.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, r≥1, {p≥2, s∈(0,N)} or {p∈[1,2),s∈((2m+2δ+1)/p,N)}, and N>5(m+δ+1). Let f^2(m) be defined by (19). Then there exists a constant C>0 such that (23)E∫ab‍f^2mx-fmx2dx≤Cln⁡nn2s/(2s+2m+2δ+1).The proof of Theorem3 is an application of a general result established by [25, Theorem 6.1]. Let us mention that ln⁡n/n2s/(2s+2m+2δ+1) corresponds to the rate of convergence obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau [15]). In the case m=0 and δ=0, this rate of convergence becomes the optimal one in the minimax sense for the standard density-regression estimation problems (see Härdle et al. [9]).In comparison to Theorem2, note that(i)
for the casep≥2 corresponding to the homogeneous zone of Besov balls (ln⁡n/n)2s/(2s+2m+2δ+1) is equal to the rate of convergence attained by f^1(m) up to a logarithmic term,(ii)
for the casep∈[1,2) corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.


## 3.2.1. Linear Wavelet Estimator

We define the linear wavelet estimatorf^1(m) by(14)f^1(m)(x)=∑k∈Λj0‍c^j0,k(m)ϕj0,k(x),x∈[a,b],where(15)c^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixXvdx,and j0 is an integer chosen a posteriori.Proposition1 presents an elementary property of c^j,k(m).Proposition 1.
Letc^j,k(m) be (15) and let cj,k(m) be (9). Suppose that(K1) holds. Then one has (16)E(c^j,k(m))=cj,k(m).Theorem2 below investigates the performance of f^1(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 2.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, p≥1, r≥1, s∈(max⁡(1/p-1/2,0),N), and N>5(m+δ+1). Let f^1(m) be defined by (14) with j0 such that(17)2j0=[n1/(2s*+2m+2δ+1)],
s
*
=
s
+
min
⁡
(
1
/
2
-
1
/
p
,
0
) ([a] denotes the integer part of a).
Then there exists a constantC>0 such that (18)E∫ab‍f^1mx-fmx2dx≤Cn-2s*/(2s*+2m+2δ+1).Note that the rate of convergencen-2s*/(2s*+2m+2δ+1) corresponds to the one obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted linear wavelet estimator (see, e.g., Chesneau [15]).The considered estimatorf^1(m) depends on s (the smoothness parameter of f(m)); it is not adaptive. This aspect, as well as the rate of convergence n-2s*/(2s*+2m+2δ+1), can be improved with thresholding methods. The next paragraph is devoted to one of them: the hard thresholding method.


## 3.2.2. Hard Thresholding Wavelet Estimator

Suppose that(K2) is satisfied. We define the hard thresholding wavelet estimator f^2(m) by(19)f^2mx=∑k∈Λτ‍c^τ,kmϕτ,kx+∑j=τj1‍∑k∈Λj‍d^j,k(m)1|d^j,k(m)|≥κλjψj,k(x),x∈[a,b], where c^j,k(m) is defined by (15),(20)d^j,k(m)=1n∑v=1n‍Yvh(Xv)12π∫-∞∞‍ixmFψj,k¯(x)F(g)(x)e-ixXvdx·1(Yv/hXv)(1/2π)∫-∞∞‍(ix)m((Fψj,k¯(x))/(F(g)(x)))e-ixXvdx≤ςj,1 is the indicator function, κ>0 is a large enough constant, j1 is the integer satisfying (21)2j1=n1/(2m+2δ+1),δ refers to (12), (22)ςj=θψ2mj2δjnln⁡n,λj=θψ2mj2δjln⁡nn,θψ=1πc2c12C12∫-∞∞‍gxdx2+E(ξ12)C12∫-∞∞‍gxdx2+E(ξ12)·∫-∞∞‍xm1+x2δFψx2dx1/2.The construction off^2(m) uses the double hard thresholding technique introduced by Delyon and Juditsky [14] and recently improved by Chaubey et al. [25]. The main interest of the thresholding using λj is to make f^2(m) adaptive; the construction (and performance) of f^2(m) does not depend on the knowledge of the smoothness of f(m). The role of the thresholding using ςj in (20) is to relax some usual restrictions on the model. To be more specific, it enables us to only suppose that ξ1 admits finite moments of order 2 (with known E(ξ12) or a known upper bound of E(ξ12)), relaxing the standard assumption E(|ξ1|k)<∞, for any k∈N.Further details on the constructions of hard thresholding wavelet estimators can be found in, for example, Donoho and Johnstone [26, 27], Donoho et al. [21, 28], Delyon and Juditsky [14], and Härdle et al. [9].Theorem3 below investigates the performance of f^2(m) in terms of rates of convergence under the MISE over Besov balls.Theorem 3.
Suppose that(K1)–(K3) are satisfied and that f(m)∈Bp,rs(M) with M>0, r≥1, {p≥2, s∈(0,N)} or {p∈[1,2),s∈((2m+2δ+1)/p,N)}, and N>5(m+δ+1). Let f^2(m) be defined by (19). Then there exists a constant C>0 such that (23)E∫ab‍f^2mx-fmx2dx≤Cln⁡nn2s/(2s+2m+2δ+1).The proof of Theorem3 is an application of a general result established by [25, Theorem 6.1]. Let us mention that ln⁡n/n2s/(2s+2m+2δ+1) corresponds to the rate of convergence obtained in the estimation of f(m) in the 1-periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau [15]). In the case m=0 and δ=0, this rate of convergence becomes the optimal one in the minimax sense for the standard density-regression estimation problems (see Härdle et al. [9]).In comparison to Theorem2, note that(i)
for the casep≥2 corresponding to the homogeneous zone of Besov balls (ln⁡n/n)2s/(2s+2m+2δ+1) is equal to the rate of convergence attained by f^1(m) up to a logarithmic term,(ii)
for the casep∈[1,2) corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.


## 3.3. Whenh Is Unknown

In the case whereh is unknown, we propose a plug-in technique which consists in estimating h in the construction of f^1(m)  (14). This yields the linear wavelet estimator f^3(m) defined by(24)f^3(m)(x)=∑k∈Λj2‍c~j2,k(m)ϕj2,k(x),x∈[a,b],where(25)c~j,km=1an∑v=1an‍Yvh^Xv1h^Xv≥c2/212π·∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixXvdx,an=[n/2], j2 is an integer chosen a posteriori, c2 refers to(K3), and h^ is an estimator of h constructed from the random variables Un=(Xan+1,…,Xn).There are numerous possibilities for the choice ofh^. For instance, h^ can be a kernel density estimator or a wavelet density estimator (see, e.g., Donoho et al. [21], Härdle et al. [9], and Juditsky and Lambert-Lacroix [29]).The estimatorf^3(m) is derived to the “NES linear wavelet estimator” introduced by Pensky and Vidakovic [16] and recently revisited in a more simple form by Chesneau [13].Theorem4 below determines an upper bound of the MISE of f^3(m).Theorem 4.
Suppose that(K1)–(K3) are satisfied, h∈L2([a*,b*]), and that f(m)∈Bp,rs(M) with M>0, p≥1, r≥1, s∈(max⁡(1/p-1/2,0),N), and N>5(m+δ+1). Let f^3(m) be defined by (24) with j2 such that 2j2≤n. Then there exists a constant C>0 such that (26)E∫ab‍f^3mx-fmx2dx≤C2(2m+2δ+1)j2max⁡E∫a*b*‍h^x-hx2dx,1n∫a*b*‍h^x-hx2dx+2-2j2s*,with s*=s+min⁡(1/2-1/p,0).The proof follows the idea of [13, Theorem 3] and uses technical operations on Fourier transforms.From Theorem4,(i)
if we choseh^=h and j2=j0 (17), we obtain Theorem 2,(ii)
ifh^ and h satisfy that there exist υ∈[0,1] and a constant C>0 such that (27)E∫a*b*‍h^x-hx2dx≤Cn-υ,then, the optimal integer j2 is such that 2j2=[nυ/(2s*+2m+2δ+1)] and we obtain the following rate of convergence for f^3(m): (28)E∫ab‍f^3mx-fmx2dx≤Cn-2s*υ/(2s*+2m+2δ+1).Naturally the estimation of h has a negative impact on the performance of f^3(m). In particular, if h∈Bp′,r′s′(M′), then the standard density linear wavelet estimator h^ attains the rate of convergence n-υ with υ=2s∘/(2s∘+1) and s∘=s′+min⁡(1/2-1/p′,0) (and it is optimal in the minimax sense for p′≥2; see Härdle et al. [9]). With this choice, the rate of convergence for f^3(m) becomes n-4s*s∘/(2s∘+1)(2s*+2m+2δ+1). Let us mention that f^3(m) is not adaptive since it depends on s. However, f^3(m) remains an acceptable first approach for the estimation of f(m) with unknown h.Conclusion and Perspectives. This study considers the estimation of f(m) from (1). According to the knowledge of h or not, we propose wavelet methods and prove that they attain fast rates of convergence under the MISE over Besov balls. Among the perspectives of this work, we retain the following.(i)
The relaxation of the assumption(K2), perhaps by considering(K2′): there exist four constants, C1>0, ω∈N, η>0, and δ≥0, such that (29)Fgx-1≤C1sinπxη-ω1+xδ,x∈R.This condition was first introduced by Delaigle and Meister [30] in a context of deconvolution-estimation of function. It implies(K2) and has the advantage to consider some functions g having zeros in Fourier transform domain as numerous kinds of compactly supported functions.(ii)
The construction of an adaptive version off^3(m) through the use of a thresholding method.(iii)
The extension of our results to theLp risk with p≥1.All these aspects need further investigations that we leave for future works.


## 4. Proofs

In this section,C denotes any constant that does not depend on j, k, or n. Its value may change from one term to another and may depend on ϕ or ψ.Proof of Proposition1.
By the independence betweenX1 and ξ1, E(ξ1)=0, sup⁡(f⋆g)=supp(h)=[a*,b*], and F(f⋆g)(x)=F(f)(x)F(g)(x), we have(30)EY1hX1e-ixX1=Ef⋆gX1hX1e-ixX1+E(ξ1)E1h(X1)e-ixX1=Ef⋆gX1hX1e-ixX1=∫a*b*‍(f⋆g)(y)h(y)e-ixyh(y)dy=∫-∞∞‍f⋆gye-ixydy=Ff⋆gx=F(f)(x)F(g)(x).
It follows from(K1) and m integration by parts that (ix)mF(f)(x)=F(f(m))(x). Using the Fubini theorem, (ix)mF(f)(x)=F(f(m))(x), (30), and the Parseval identity, we obtain (31)Ec^j,km=EY1h(X1)12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)e-ixX1dx=12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)EY1h(X1)e-ixX1dx=12π∫-∞∞‍ixmFϕj,k¯(x)F(g)(x)F(f)(x)F(g)(x)dx=12π∫-∞∞‍ixmF(f)(x)Fϕj,k¯(x)dx=12π∫-∞∞‍FfmxFϕj,k¯xdx=∫ab‍fm(x)ϕj,k(x)dx=cj,k(m).
Proposition1 is proved.Proof of Theorem2.
We expand the functionf(m) on B as (8) at the level l=j0. Since B forms an orthonormal basis of L2([a,b]), we get(32)E∫ab‍f^1mx-fmx2dx=∑k∈Λj0‍Ec^j0,km-cj0,km2+∑j=j0∞‍∑k∈Λj‍dj,km2.Using Proposition 1, (Y1,X1),…,(Yn,Xn) that are i.i.d., the inequalities V(D)≤E(|D|2) for any random complex variable D and (x+y)2≤2(x2+y2), (x,y)∈R2, and(K1) and(K3), we have(33)Ec^j0,km-cj0,km2=Vc^j0,k(m)=1nVY1hX112π∫-∞∞‍ixmFϕj0,k¯xFgxe-ixX1dx≤12π2nEY1hX1∫-∞∞‍xmFϕj0,k¯xFgxe-ixX1dx2≤22π2nEf⋆gX12+ξ12hX12∫-∞∞‍xmFϕj0,k¯xFgxe-ixX1dx2·∫-∞∞‍xmFϕj0,k¯xFgxe-ixX1dx2≤1n22π2c2C12∫-∞∞‍gxdx2+Eξ12·E1hX1∫-∞∞‍xmFϕj0,k¯xFgxe-ixX1dx2.
The Parseval identity yields(34)E1h(X1)∫-∞∞‍xmFϕj0,k¯(x)F(g)(x)e-ixX1dx2=∫a*b*‍1h(y)∫-∞∞‍xmFϕj0,k¯(x)F(g)(x)e-ixydx2h(y)dy≤∫-∞∞‍FxmFϕj0,k¯xFgxy2dy=2π∫-∞∞‍xmFϕj0,k¯(x)F(g)(x)2dx.Using(K2), Fϕj0,k(x)=2-j0/2Fϕ(x/2j0), and a change of variables, we obtain(35)∫-∞∞‍xmFϕj0,k¯xFgx2dx≤1c12∫-∞∞‍x2m1+x2δFϕj0,k(x)2dx=1c122-j0∫-∞∞‍x2m1+x2δFϕx2j02dx=1c12∫-∞∞‍22j0mx2m1+22j0x2δFϕ(x)2dx≤1c122(2m+2δ)j0∫-∞∞‍x2m1+x2δFϕ(x)2dx≤C2(2m+2δ)j0.(Let us mention that ∫-∞∞‍x2m(1+x2)δFϕ(x)2dx is finite thanks to N>5(m+δ+1).)
Combining (33), (34), and (35), we have (36)Ec^j0,km-cj0,km2≤C2(2m+2δ)j01n.For the integer j0 satisfying (17), it holds that(37)∑k∈Λj0‍Ec^j0,km-cj0,km2≤C22m+2δ+1j01n≤Cn-2s*/(2s*+2m+2δ+1).Let us now bound the last term in (32). Since f(m)∈Bp,rs(M)⊆B2,∞s*(M) (see [9, Corollary 9.2]), we obtain(38)∑j=j0∞‍∑k∈Λj‍dj,km2≤C2-2j0s*≤Cn-2s*/(2s*+2m+2δ+1).Owing to (32), (37), and (38), we have (39)E∫ab‍f^1mx-fmx2dx≤Cn-2s*/(2s*+2m+2δ+1).Theorem 2 is proved.Proof of Theorem3.
Forγ∈{ϕ,ψ}, any integer j≥τ, and k∈Λj, (a1)
using arguments similar to those in Proposition1, we obtain (40)E1n∑v=1n‍YvhXv12π∫-∞∞‍ixmFγj,k¯xFgxe-ixXvdx=∫ab‍fmxγj,kxdx,(a2)
using (33), (34), and (35) with γ instead of ϕ, we have (41)∑v=1n‍EYvh(Xv)12π∫-∞∞‍ixmFγj,k¯(x)F(g)(x)e-ixXvdx2=nEY1hX112π∫-∞∞‍xmFγj,k¯xFgxe-ixX1dx2≤C*2n2(2m+2δ)j,with C*2=(1/(πc2c12))(C12(∫-∞∞‍|g(x)|dx)2+E(ξ12))∫-∞∞‍xm(1+x2)δ|F(γ)(x)|2dx.Thanks to(a1) and(a2), we can apply [25, Theorem 6.1] (see Appendix) with μn=υn=n, σ=m+δ, θγ=C*, Wv=(Yv,Xv), (42)qv(γ,(y,z))=yh(z)12π∫-∞∞‍ixmFγj,k¯(x)F(g)(x)e-ixzdxand f(m)∈Bp,rs(M) with M>0, r≥1, either {p≥2 and s∈(0,N)} or {p∈[1,2) and s∈(1/p,N)}, we prove the existence of a constant C>0 such that(43)E∫ab‍f^2mx-fmx2dx≤Cln⁡nn2s/(2s+2m+2δ+1).Theorem 3 is proved.Proof of Theorem4.
We expand the functionf(m) on B as (8) at the level l=j2. Since B forms an orthonormal basis of L2([a,b]), we get(44)E∫ab‍f^3mx-fmx2dx=∑k∈Λj2‍Ec~j2,km-cj2,km2+∑j=j2∞‍∑k∈Λj‍dj,km2.Using f(m)∈Bp,rs(M)⊆B2,∞s*(M) (see [9, Corollary 9.2]), we have(45)∑j=j2∞‍∑k∈Λj‍dj,km2≤C2-2j2s*.Let c^j2,k(m) be (15) with n=an and j=j2. The elementary inequality (x+y)2≤2(x2+y2), (x,y)∈R2, yields(46)∑k∈Λj2‍Ec~j2,km-cj2,km2≤2(S1+S2),where (47)S1=∑k∈Λj2‍Ec~j2,km-c^j2,km2,S2=∑k∈Λj2‍Ec^j2,km-cj2,km2.
Upper Bound for
S
2
. Proceeding as in (37), we get(48)S2≤C2(2m+2δ+1)j21an≤C2(2m+2δ+1)j21n.
Upper Bound for
S
1
. The triangular inequality gives (49)c~j2,km-c^j2,km≤12πan∑v=1an‍Yv∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixXvdx·1h^(Xv)1|h^(Xv)|≥c2/2-1h(Xv).Owing to the triangular inequality, the indicator function,(K3), |h^(Xv)|<c2/2⊆|h^(Xv)-h(Xv)|>c2/2, and the Markov inequality, we have (50)1h^(Xv)1|h^(Xv)|≥c2/2-1h(Xv)=1h(Xv)h(Xv)-h^(Xv)h^(Xv)1|h^(Xv)|≥c2/2-h(Xv)-h^(Xv)h^(Xv)1|h^(Xv)|≥c2/21|h^(Xv)|<c2/2≤1hXv2c2h^Xv-hXv+1|h^(Xv)-h(Xv)|>c2/2≤4c2h^Xv-hXvhXv.Therefore (51)c~j2,k(m)-c^j2,k(m)≤CAj2,k,n,where (52)Aj,k,n=1an∑v=1an‍Yv∫-∞∞‍xmFϕj2,k¯xFgxe-ixXvdx·|h^(Xv)-h(Xv)|h(Xv).Let us now consider Un=(Xan+1,…,Xn). For any complex random variable D, we have the equality (53)ED2=EED2∣Un=EVD∣Un+EED∣Un2,where E(D∣Un) denotes the expectation of D conditionally to Un and V(D∣Un)  and the variance of D conditionally to Un. Therefore(54)S1≤C∑k∈Λj2‍E(Aj2,k,n2)=C(Wj2,n+Zj2,n),where (55)Wj2,n=∑k∈Λj2‍EVAj2,k,n∣Un,Zj2,n=∑k∈Λj2‍EEAj2,k,n∣Un2.Let us now observe that, owing to the independence of (Y1,X1),…,(Yn,Xn), the random variables |Y1|∫-∞∞‍xm(Fϕj2,k¯(x)/F(g)(x))e-ixX1dx|h^(X1)-h(X1)|/h(X1),…, |Yan|∫-∞∞‍xm(Fϕj2,k¯(x)/F(g)(x))e-ixXandx|h^(Xan) − h(Xan)|/h(Xan) conditionally to Un are independent. Using this property with the inequalities V(D∣Un)≤E(D2∣Un) for any complex random variable D and (x+y)2≤2(x2+y2), (x,y)∈R2, the independence between X1 and ξ1,(K1) and(K3), we get (56)VAj2,k,n∣Un=1anVY1∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx·h^X1-hX1h(X1)∣Un≤1anEY12∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx2∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx2·h^(X1)-h(X1)h(X1)2∣Un≤1an2c2C12∫-∞∞‍gxdx2+E(ξ12)·E∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx2∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx2·h^X1-hX12h(X1)∣Un=2c2C12∫-∞∞‍gxdx2+E(ξ12)·1an∫a*b*‍∫-∞∞‍xmFϕj2,k¯xFgxe-ixydx2·h^y-hy2h(y)h(y)dy≤C1n∫a*b*‍∫-∞∞‍xmFϕj2,k¯xFgxe-ixydx2·h^y-hy2dy.Owing to(K2), Fϕj2,k(x)=2-j2/2Fϕ(x/2j2), and a change of variables, we obtain (57)∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixydx≤∫-∞∞‍xmFϕj2,kxFgxdx≤1c1∫-∞∞‍xm1+x2δ/2Fϕj2,k(x)dx=1c12-j2/2∫-∞∞‍xm1+x2δ/2Fϕx2j2dx=1c12j2/2∫-∞∞‍2j2mxm1+22j2x2δ/2Fϕ(x)dx≤1c12(m+δ+1/2)j2∫-∞∞‍xm1+x2δ/2Fϕ(x)dx≤C2(m+δ+1/2)j2.
Therefore, usingCard(Λj2)≤C2j2 and 2j2≤n, we obtain(58)Wj2,n≤C2(2m+2δ+1)j22j21nE∫a*b*‍h^y-hy2dy≤C2(2m+2δ+1)j2E∫a*b*‍h^y-hy2dy.Now, by the Hölder inequality for conditional expectations, arguments similar to (33), (34), and (35), we get (59)EAj2,k,n∣Un=E|Y1|∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx·|h^(X1)-h(X1)|h(X1)∣Un≤EY12h(X1)∫-∞∞‍xmFϕj2,k¯xFgxe-ixX1dx2·∫-∞∞‍xmFϕj2,k¯xFgxe-ixX1dx2∣Un1/2·Eh^X1-hX12h(X1)∣Un1/2=EY12h(X1)∫-∞∞‍xmFϕj2,k¯(x)F(g)(x)e-ixX1dx21/2·∫a*b*‍h^y-hy2h(y)h(y)dy1/2≤C2(m+δ)j2∫a*b*‍h^y-hy2dy1/2.Hence(60)Zj2,n≤C2(2m+2δ+1)j2E∫a*b*‍h^y-hy2dy.It follows from (54), (58), and (60) that(61)S1≤C2(2m+2δ+1)j2E∫a*b*‍h^y-hy2dy.Putting (46), (48), and (61) together, we get(62)∑k∈Λj2‍Ec~j2,km-cj2,km2≤C2(2m+2δ+1)j2max⁡E∫a*b*‍h^y-hy2dy,1n.
Combining (44), (45), and (62), we obtain the desired result; that is, (63)E∫ab‍f^3mx-fmx2dx≤C22m+2δ+1j2max⁡E∫a*b*‍h^y-hy2dy,1n+E∫a*b*‍h^y-hy2dy,1n2-2j2s*.Theorem 4 is proved.



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*Source: 695904-2015-03-22.xml*