# Wavelet Estimation of Function Derivatives from a Multichannel Deconvolution Model

Authors: Huijun Guo; Shuzi Li
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper considers a multichannel deconvolution model with Gaussian white noises. The goal is to estimate thed-th derivatives of an unknown function in the model. For super-smooth case, we construct an adaptive linear wavelet estimator by wavelet projection method. For regular-smooth case, we provide an adaptive nonlinear wavelet estimator by hard-thresholded method. In order to measure the global performances of our estimators, we show upper bounds on convergence rates using the Lp-risk (1≤p<∞).


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## Body

## 1. Introduction

We consider a multichannel deconvolution problem with Gaussian white noises. The signalf∈L2T (T=0,1) is observed indirectly in noisy signals,
(1)dYℓt=f∗gℓtdt+σℓn1/2dBℓt,t∈T,ℓ=1,2,⋯,M,whereσℓ are known positive constants and M is the number of channel. Suppose f,gℓ∈L2T are 1-periodic on ℝ, the blurring functions gℓ are known and
(2)f∗gℓt=∫Tgℓt−xfxdx,t∈T.Additive noises are present withBℓ· (ℓ=1,2,⋯,M) denoting independent standard Brownian motions. The goal is to estimate the derivatives fdd∈ℕ,f0=f of the signal f from the data Yℓtℓ=1,2,⋯,M.The notationA≲B indicates that A≤cB with a positive constant c which is independent of A and B; A≳B means B≲A; A≍B stands for A≲B and B≲A.Forh∈L2T, its Fourier coefficients are defined by
(3)hω=∫Te−2πiωthtdt,ω∈ℤ.In the Fourier domain, the smooth-type convolution functions are of the form(4)gω,ℓ≍1+ω2−vℓ/2exp−θℓωβℓ,whereω∈ℤ, ℓ=1,2,⋯,M; βℓ>0 and θℓ≥0. The so-called super-smooth convolution or exponential decay occurs when θℓ>0,vℓ∈ℝ (ℓ=1,2,⋯,M) and the regular-smooth convolution or polynomial decay occurs when θℓ=0,vℓ>0 (ℓ=1,2,⋯,M).There are many works on deconvolution Gaussian white noise model, see [1–5]. In the single-channel deconvolution case (M=1), [6] constructs a kernel estimator and obtains the optimal convergence rates over the Sobolev spaces. Furthermore, [7, 8] provide the wavelet estimation with a general noise assumption. In the multichannel deconvolution case (M>1), [9] considers the model in a periodic setting, proposes adaptive wavelet-thresholding estimators by weight method, and obtains the upper bounds on Lp-risk (1<p<∞). Moreover, [10] generalizes the results in [9] by relaxing the noise condition. As an example of a multichannel deconvolution model, multiple light detection and ranging (LIDAR) devices are used to receive the return signals to determine the distance and location of the reflecting material (see [10]). The effective accuracy loss caused by radar signal propagation can be corrected by deconvolution.We focus on a more general problem: estimate thed-th derivative fd of d∈ℕ,f0=f. This is of interest to detect possible bumps, concavity, or convexity properties of f. The problem of estimating derivatives arises in many scientific settings (see [11–15]), such as signal processing, chemistry, and geophysics. For example, higher-order derivatives can detect important features in restoring distorted and noise-containing digital images. These features can be used to reconstruct and restore digital images, see [13]. Derivatives can be used for raw spectral detection in the field of chemistry, see [11]. Evaluation of higher derivatives (gradients) of potential fields plays an important role in geophysical interpretation, see [12].Derivative estimation has been studied in many statistical model, see [16–21]. For example, [17] develops an adaptive estimator of d-th derivatives of unknown function in the standard deconvolution model (M=1) and proves that it achieves near optimal rates of convergence under the mean integrated squared error (MISE) over a wide range of smoothness classes. [18] studies the heteroscedastic multichannel deconvolution model. The number of channels is equal to the number of samples, which tends to infinity in the model. Under the assumption that the Fourier decay of the blurring functions is at a same polynomial rate in regular-smooth case, they construct the adaptive wavelet block-thresholding estimator for the d-th derivative function and measure the estimation accuracy by MISE over Besov spaces. Recently, [20] generalizes the results of [18] from L2- to Lp- (1≤p<∞) risk case.Wavelet method has been a popular choice in deconvolution problems due to their local property in both time and frequency domain, see [22, 23]. Motivated by [9, 18, 20], we will construct the wavelet estimator of fd (d≥1) from model (1) and examine the estimation accuracy by Lp-risk (1≤p<∞).The paper is organized in the following way. Section2 introduces the periodized Meyer wavelets and Besov spaces. We construct the wavelet estimators for the d-th derivative fd in Section 3. The convergence rates of linear and nonlinear estimators are given in Section 4 and Section 5, respectively. For both super-smooth and regular-smooth case, we obtain the adaptive wavelet estimators and measure their performances by Lp-risk (1≤p<∞). The last part provides some conclusions.


## 2. Wavelets and Besov Space

In order to establish the convergence rates, we use the periodized Meyer’s wavelet basis on the unit interval. Letφ· and ψ· be the Meyer scaling and wavelet functions (see [24]), respectively. As usual, for any j∈ℤ, k∈ℤ,
(5)φj,kt=2j/2φ2jt−k,ψj,kt=2j/2ψ2jt−k,t∈ℝ.The periodic scaling and wavelet functions are defined by(6)Φj,kt=∑l∈ℤφj,kt−l,Ψj,kt=∑l∈ℤψj,kt−l,t∈T,with j≥0 and k=0,1,⋯,2j−1. For an integer j0≥0, the collection
(7)Φj0,k·:k=0,1,⋯,2j0−1∪Ψj,k·:j≥j0,k=0,1,⋯,2j−1,constitutes an orthonormal basis ofL2T (see [25]).In the periodic setting, Besov spaces can be defined by the behavior of the wavelet coefficients (see [26]). Denote the inner product operator, h1,h2=∫Th1th2¯tdt, where h¯ denotes the complex conjugate of h.Definition 1.
The periodic Besov spaces are defined via periodic wavelets,(8)Br,qsT=h∈LrT:hBr,qs<∞,1≤r,q<∞,where
(9)hBr,qs=h,Φ0,0r1/r+∑j=0∞2qjs+1/2−1/r∑k=02j−1h,Ψj,krq/r1/q.
By the above definition, the following embedding result can be easily obtained:(10)Br,qsT⊂Br′,qs′T,s,s′>0,r≤r′ands′=s−1r+1r′.


## 3. Wavelet Estimators

The estimation is provided in the Fourier domain to reduce the convolution operator to a product of Fourier coefficients.Let the Fourier basis functionseωt=e2πiωt, ω∈ℤ. Let hℓ=f∗gℓ. Denote the relevant Fourier coefficients,
(11)yω,ℓ=∫Teω¯tdYℓt,hω,ℓ=hℓ,eω,zω,ℓ=∫Teω¯tdBℓt,(12)fdω=fd,eω,gω,ℓ=gℓ,eω,d∈ℕ,ℓ=1,2,⋯,M.We consider model (1) in the Fourier domain. Then,
(13)yω,ℓ=hω,ℓ+σℓn1/2zω,ℓ,ω∈ℤ,ℓ=1,2,⋯,M,(14)hω,ℓ=gω,ℓfω,ω∈ℤ,ℓ=1,2,⋯,M.Following a similar procedure to [9], (14) leads to the following expression for Fourier coefficients of the target function,
(15)fω=∑ℓ=1Mγℓgω,ℓ¯hω,ℓ∑ℓ=1Mγℓgω,ℓ2,ω∈ℤ,whereγℓ are weights to be specified later.Assume thatfd∈L2T. Then fd can be expanded into a wavelet series as
(16)fdt=∑k=02j0−1aj0,kΦj0,kt+∑j=j0∞∑k=02j−1bj,kΨj,kt,t∈T,where
(17)aj0,k=fd,Φj0,k,bj,k=fd,Ψj,k.Define the following collection,(18)Djε=ω∈ℤ:ω≤2j+13,ε=0,ω∈ℤ:2j3≤ω≤2j+23,ε=1.Let us now investigate the estimation ofbj,k. Assume that f is d-times (d∈ℕ) differentiable on ℝ and fd∈L2T. Then, for any d∈ℕ, fdω=2πiωdfω. This with the Parseval identity shows
(19)bj,k=∫TfdtΨj,ktdt=∑ω∈Dj1fdωΨωj,k¯=∑ω∈Dj12πiωdfωΨωj,k¯=∑ω∈Dj12πiωd∑ℓ=1Mγℓgω,ℓ¯hω,ℓ∑ℓ=1Mγℓgω,ℓ2Ψωj,k¯.According to (13), the wavelet coefficient can be estimated with
(20)b^j,k=∑ω∈Dj12πiωd∑ℓ=1Mγℓgω,ℓ¯yω,ℓ∑ℓ=1Mγℓgω,ℓ2Ψωj,k¯.Similarly,a^j0,k are defined by
(21)a^j0,k=∑ω∈Dj002πiωd∑ℓ=1Mγℓgω,ℓ¯yω,ℓ∑ℓ=1Mγℓgω,ℓ2Φωj0,k¯.We now present the considered procedure for the estimation offd. Denote the parameter ℓ∗=argmin1≤ℓ≤M2vℓeθℓ4/3βℓ in the following.The linear wavelet estimatorf^n,dlind≥0 is defined by
(22)f^n,dlint=∑k=02j0−1a^j0,kΦj0,kt,where
(23)2j0≍n1/2s″+2vℓ∗+2d+1s″=s−1minr,p+1p, in the case of regular-smooth convolution and
(24)2j0≍1−ϵlnn2θℓ∗2/3βℓ∗1/βℓ∗0<ϵ<1,in the case of super-smooth convolution.In the regular-smooth case,2j0 depends on smooth parameter s, which leads to the linear estimator nonadaptive. Hence, we construct the following adaptive nonlinear wavelet estimator f^n,dnon by hard-thresholded method,
(25)f^n,dnont=∑k=02j0−1a^j0,kΦj0,kt+∑j=j0j1∑k=02j−1b^j,k1b^j,k≥λjΨj,kt,where
(26)2j0≍lnn,2j1≍n1/2vℓ∗+2d+1,and1A denotes the indicator function of the set A. Here, the parameters λj are given by
(27)λj=ρχjjn,where
(28)ρ>0,(29)χj2=∑ω∈Dj12πω2dΨωj,k2∑ℓ=1Mσℓ−2gω,ℓ2−1.


## 4. Convergence Rates of the Linear Estimators

We will show the convergence rates of the linear wavelet estimator for regular-smooth and super-smooth case, respectively.Now, we need to clarify two lemmas, which will be used in later discussions.Lemma 2.
Consider model (1) with Condition (4). Suppose f is d-times d∈ℕ differentiable on ℝ and fd∈L2T. If a^j0,k and b^j,k are defined as (21) and (20), respectively, then
(i)
Thea^j0,k is complex Gaussian with(30)Ea^j0,k=aj0,k,Vara^j0,k=∑ω∈Dj002πω2dΦωj0,k2∑ℓ=1Mσℓ−2ngω,ℓ2−1.
Furthermore,Rea^j0,k is independent of Ima^j0,k, ERea^j0,k=EIma^j0,k and VarRea^j0,k=VarIma^j0,k;
(ii)
Theb^j,k is complex Gaussian with(31)Eb^j,k=bj,k,Varb^j,k=∑ω∈Dj12πω2dΨωj,k2∑l=1Mσℓ−2ngω,ℓ2−1.Similarly,Reb^j,k is independent of Imb^j,k, EReb^j,k=EImb^j,k, and VarReb^j,k=VarImb^j,k.Proof.
We just show Lemma2 (ii) is true. The Conclusion (i) can be proved similarly.
According to Formulas (13), (19), and (20), we obtain
(32)b^j,k=bj,k+∑ω∈Dj12πiωd∑ℓ=1Mγℓgω,ℓ¯σℓn−1/2zω,ℓ∑ℓ=1Mγℓgω,ℓ2Ψωj,k¯.
The noise termszω,ℓ in (11) are complex Gaussian with Ezω,ℓ=0. Hence,
(33)Eb^j,k=bj,k,
which meansb^j,k is unbias.
For eachℓ=1,2,⋯,M,
(34)Ezω,ℓzη,ℓ¯=E∫Teωt¯dBℓt∫Teηt′¯dBℓt′¯=∫Teωt¯eηtdt=∫Te−2πitω−ηdt=δω,η,where δa,b is the Kronecker delta function. Since Bℓt is independent of Bℓ′t for ℓ≠ℓ′, Ezω,ℓzω′,ℓ′¯=0. Thus, for each ω,ω′∈ℤ2, ℓ,ℓ′∈1,2,⋯,M2, we obtain
(35)Ezω,ℓzω′,ℓ′¯=1,ω=ω′,ℓ=ℓ′,0,otherwise.
Now, we compute the variance ofb^j,k. Hence,
(36)Varb^j,k=E∑ω∈Dj12πiωd∑ℓ=1Mγℓgω,ℓ¯σℓn−1/2zω,ℓ∑ℓ=1Mγℓgω,ℓ2Ψωj,k¯∑ω∈Dj12πiωd∑ℓ=1Mγℓgω,ℓ¯σℓn−1/2zω,ℓ∑ℓ=1Mγω,ℓgω,ℓ2Ψωj,k¯¯=∑ω∈Dj12πω2d∑ℓ=1Mγℓ2σℓ2n−1gω,ℓ2Ψωj,k2∑ℓ=1Mγℓgω,ℓ22.
By Cauchy-Schwarz inequality, one can obtain(37)∑ℓ=1Mγℓgω,ℓ22≤∑ℓ=1Mγℓ2σℓ2n−1gω,ℓ2∑ℓ=1Mσℓ−2ngω,ℓ2,where the equality only holds for γℓ=nσℓ−2. Using the optimal weights γℓ=nσℓ−2, one has
(38)Varb^j,k=∑ω∈Dj12πω2dΨωj,k2∑ℓ=1Mσℓ−2ngω,ℓ2−1.
By [27], zω,ℓω∈Dj1 are independent and identically distributed complex Gaussian random variables with mean zero and unit variance. Hence, b^j,k is complex Gaussian random. It is easy to obtain that Reb^j,k is independent of Imb^j,k, EReb^j,k=EImb^j,k and VarReb^j,k=VarImb^j,k.Remark 3.
The channel numberM is fixed, while those of [18, 20] tend to infinity. To obtain Lemma 2, we construct the wavelet coefficient estimators (20) and (21) by weighted summation method. The weights γℓ are determined by Cauchy-Schwarz inequality.Lemma 4.
Consider model (1) with Condition (4). Suppose f is d-times d∈ℕ differentiable on ℝ and fd∈L2T. Then for any 1≤p<∞,
(i)
Super-smooth case(39)Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+depθℓ∗2j0+13βℓ∗,(ii)
Regular-smooth case(40)Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+d,Eb∧j,k−bj,kp≲n−p/22pjvℓ∗+d.Proof.
We just prove the conclusions ona^j0,k. The result on b^j,k can be obtained similarly. For super-smooth case, we obtain
(41)Vara^j0,k≤supω∈Dj002πω2d∑ℓ=1Mσℓ−2ngω,ℓ2−1∑ω∈Dj00Φωj0,k2,
by Lemma2 (i). It follows from Parseval identity and ∫TΦj0,kt2dt=1 that ∑ω∈Dj00Φωj0,k2=1. Hence,
(42)Vara^j0,k≲supω∈Dj002πω2d∑ℓ=1Mσℓ−2ngω,ℓ2−1.
According to (4) and Dj00=ω∈ℤ,ω≤2j0+1/3,
(43)Vara^j0,k≲22dj0∑ℓ=1Mσℓ−2ninfω≤2j0+1/31+ω2−vℓexp−2θℓωβℓ−1≲22dj0∑ℓ=1Mσℓ−2n2−2j0vℓexp−2θℓ2j0+13βℓ−1≲n−122j0vℓ∗+dexp2θℓ∗2j0+13βℓ∗.
DefineUj0,k=a^j0,k−aj0,k. By Lemma 2 (i), Uj0,k is complex Gaussian with EUj0,k=0 and
(44)VarUj0,k=∑ω∈Dj002πω2dΦωj0,k2∑ℓ=1Mσℓ−2ngω,ℓ2−1.
Furthermore,ReUj0,k is independent of ImUj0,k, EReUj0,k=EImUj0,k, and VarReUj0,k=VarImUj0,k. Hence,
(45)EUj0,kp=CpVarUj0,kp/2,where Cp=2∫0+∞xp+1e−x2dx. By Uj0,k=a^j0,k−aj0,k and (43), one has
(46)Ea∧j0,k−aj0,kp≲Vara∧j0,kp/2≲n−p/22pj0vℓ∗+depθℓ∗2j0+13βℓ∗.
The regular-smooth case leads toθℓ∗=0. Hence,
(47)Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+d,for the regular-smooth case.Remark 5.
In our discussion, we choose parameterℓ∗=argmin1≤ℓ≤M2vℓeθℓ4/3βℓ. Now, we explain the regular-smooth case. Our Fourier decay of the blurring function gℓ is at a polynomial rate vℓ, which changes with the channel index ℓ. The regular-smooth case implies ℓ∗=argmin1≤ℓ≤M2vℓ, which stands for the optimal channel. However, [18, 20] assume that the Fourier decay of the blurring functions is at a same polynomial rate in regular-smooth case. If vℓ1≤ℓ≤M are same, our condition of blurring functions reduces to [18, 20].Now, we are in the position to state the main result of this section.Theorem 6.
Consider model (1) with Condition (4). Suppose f is d-times d∈ℕ differentiable on ℝ, fd∈L2T∩Br,qsT1≤r,q<∞,s≥1/r, and 1≤p<∞.The linear wavelet estimator f^n,dlin is defined by (22).
In the case of regular-smooth convolution,f^n,dlin with
(48)2j0≍n1/2s″+2vℓ∗+2d+1,satisfies
(49)Ef^n,dlin−fpp≲n−s″p/2s″+2vℓ∗+2d+1.
In the case of super-smooth convolution,f^n,dlin with
(50)2j0≍1−ϵlnn2θℓ∗2/3βℓ∗1/βℓ∗0<ϵ<1,satisfies
(51)Ef^n,dlin−fpp≲lnn−s′′p/βℓ∗.Here,βℓ∗>0, s′′=s−1/minr,p+1/p.Proof.
By (16) and (22), we get
(52)Ef^n,dlin−fpp≲E1+E2,where
(53)E1=E∑k=02j0−1a^j0,k−aj0,kΦj0,kpp,E2=∑j≥j0∑k=02j−1bj,kΨj,kpp.
Firstly, we consider the regular-smooth case.
Estimate the termE1. By Lemma 4 and (48), one has
(54)E1≲2j0p/2−1∑k=02j0−1Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+d+1/2≲n−s″p/2s″+2vℓ∗+2d+1.
Estimate the termE2. For r≤p, we obtain
(55)E2≲∑j≥j0∑k=02j−1bj,kΨj,kpp≲∑j≥j02−js′2js′+1/2−1/p∑k=02j−1bj,kp1/pp,
withs′=s−1/r+1/p. The Besov spaces embedding result (10) implies fd∈Br,qsT⊆Bp,qs′T. By the definition of Besov spaces,
(56)E2≲∑j≥j02−js′p≲2−j0ps′.
Forr>p, one gets
(57)E2≤∑j≥j0∑k=02j−1bj,kΨj,krrp/r≲2−j0rsp/r=2−j0sp,by Hölder inequality and (56) with p=r. The above inequality with (56) shows
(58)E2≲2−j0ps+1/p−1/minp,r=2−j0ps″,holds for 1≤p<∞. By (48), we can obtain
(59)E2≲2−j0ps″≲n−s″p/2s″+2vℓ∗+2d+1.
Hence, (53), (54), and the above inequality lead to
(60)Ef^n,dlin−fpp≲E1+E2≲n−s″p/2s″+2vℓ∗+2d+1,for 1≤p<∞.
Nextly, we consider the super-smooth case.
Estimate the termE1. By Lemma 4 and (50), we have
(61)E1≲2j0p/2−1∑k=02j0−1Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+d+1/2epθℓ∗2j0+1/3βℓ∗≲n−εp/2lnnpvℓ∗+d+1/2/βℓ∗=olnn−s″p/βℓ∗,where s″=s−1/minr,p+1/p.
Estimate the termE2. Similar to the regular-smooth case,
(62)E2≲2−j0ps″≲lnn−s″p/βℓ∗.
This with (53) and (61) shows
(63)Ef^n,dlin−fpp≲lnn−s′′p/βℓ∗,for 1≤p<∞.Remark 7.
Theorem6 shows that the regular-smooth case has a better convergence rate than the super-smooth case. The convergence rate of regular-smooth case depends on d, while that of super-smooth case does not. In the super-smooth case, (50) implies 2j0 does not depend on the smooth parameter s, which shows the linear wavelet estimator adaptive. However, the linear wavelet estimator in the regular-smooth case is nonadaptive for (48) depending on s. We will construct adaptive wavelet estimator for the regular-smooth case in later section.


## 5. Convergence Rates of the Nonlinear Estimator

In this section, we will give the convergence rates of nonlinear wavelet estimator for regular-smooth case. We provide a useful lemma at the beginning.Lemma 8.
Consider model (1) with Condition (4). Suppose f is d-times d∈ℕ differentiable on ℝ and fd∈L2T. If λj is defined as (27), then
(64)ℙb^j,k−bj,k≥λj2≲2−ρ2j/4.Proof.
DenoteVj,k=b^j,k−bj,k. By Lemma 2 (ii), Vj,k is complex Gaussian with EVj,k=0 and VarVj,k=n−1χj2. Furthermore, ReVj,k is independent of ImVj,k, EReVj,k=EImVj,k, and VarReVj,k=VarImVj,k. Obviously,
(65)ℙVj,kVarVj,k≥λ=e−λ2.
Hence,(66)ℙb^j,k−bj,k≥λj2=ℙb^j,k−bj,k≥ρχjj/n2=ℙb^j,k−bj,kχj/n≥ρj2=e−ρ2j/4≤2−ρ2j/4.Theorem 9.
Consider model (1) with Condition (4). Suppose f is d-times d∈ℕ differentiable on ℝ, fd∈L2T∩Br,qsT1≤r,q<∞,s≥1/r, and 1≤p<∞. For the case of regular-smooth convolution, the nonlinear estimator f^n,dnon defined by (25)–(28) satisfies
(67)Ef^n,dnon−fpp≲lnn3p/2n−sp/2s+2vℓ∗+2d+1,s≥2vℓ∗+2d+12pr−1,lnn3p/2n−s′p/2s−1/r+2vℓ∗+2d+1,s<2vℓ∗+2d+12pr−1,where s′=s−1/r+1/p.Proof.
Whenr>p, the space LrT is continuously embedded into the space LpT, then the Jensen’s inequality implies
(68)Ef^n,dnon−fpp≲Ef^n,dnon−frp≤Ef∧n,dnon−frrp/r.
Therefore, we only need to show the conclusion forr≤p.
By (16) and (25), we obtain
(69)Ef^n,dnon−fpp≲E3+E4+E5,where
(70)E3=E∑k=02j0−1a^j0,k−aj0,kΦj0,kpp,E4=E∑j=j0j1∑k=02j−1b^j,k1b^j,k≥λj−bj,kΨj,kpp,E5=∑j>j1∑k=02j−1bj,kΨj,kpp.
According to Lemma4 (ii), we derive that
(71)E3≲2j0p/2−1∑k=02j0−1Ea∧j0,k−aj0,kp≲n−p/22pj0vℓ∗+d+1/2.
Choose(72)2j0∗≍n1−2s∗/2vℓ∗+2d+1,where s∗=mins/2s+2vℓ∗+2d+1,s′/2s−1/r+2vℓ∗+2d+1. Then 2j0≲2j0∗, and
(73)E3≲n−p/22pj0∗vℓ∗+d+1/2≲n−s∗p≲n−sp/2s+2vℓ∗+2d+1,s≥2vℓ∗+2d+12pr−1,n−s′p/2s−1/r+2vℓ∗+2d+1,s<2vℓ∗+2d+12pr−1.
Similar to (55) and (56), we obtain
(74)E5≲2−j1s′p,
withs′=s−1/r+1/p. Taking 2j1∗≍minns/s′2s+2vℓ∗+2d+1,n1/2s−1/r+2vℓ∗+2d+1, then 2j1∗≲2j1. This leads to
(75)E5≲2−j1∗s′p≲n−sp/2s+2vℓ∗+2d+1,s≥2vℓ∗+2d+12pr−1,n−s′p/2s−1/r+2vℓ∗+2d+1,s<2vℓ∗+2d+12pr−1.
Finally, we consider the term(76)E4=E∑j=j0j1∑k=02j−1b^j,k1b^j,k≥λj−bj,kΨj,kpp.
Utilizing properties of convex functions andj1−j0≲lnn, we obtain that
(77)E4≲j1−j0+1p−1∑j=j0j1E∑k=02j−1b^j,k1b^j,k≥λj−bj,kΨj,kpp≲j1−j0+1p−1∑j=j0j12jp/2−1∑k=02j−1Eb∧j,k1b∧j,k≥λj−bj,kp≲lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Eb∧j,k1b∧j,k≥λj−bj,kp.
The definition ofλj is shown in (27). Define Aj=k:b^j,k≥λj, Bj=k:bj,k≥λj/2, Cj=k:bj,k≥2λj. Then
(78)∑k=02j−1Eb∧j,k1b∧j,k≥λj−bj,kp=∑k=02j−1Eb∧j,k−bj,kpIk∈Aj∩Bj+Ik∈Aj∩Bjc+∑k=02j−1Ebj,kpIk∈Ajc∩Cj+Ik∈Ajc∩Cjc.
Therefore, we need to estimate the following items:(79)E4≲E41+E42+E43+E44,where
(80)E41=lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Eb∧j,k−bj,kpIk∈Aj∩Bj,E42=lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Eb∧j,k−bj,kpIk∈Aj∩Bjc,E43=lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Ebj,kpIk∈Ajc∩Cj,E44=lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Ebj,kpIk∈Ajc∩Cjc.
ObviouslyAj∩Bjc⊂Dj, Ajc∩Cj⊂Dj, where Dj=k:b^j,k−bj,k≥λj/2. Meanwhile, bj,k<2b^j,k−bj,k for b^j,k<λj and bj,k≥2λj. Hence,
(81)E42+E43≲lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Eb∧j,k−bj,kpIk∈Dj.
This with Hölder inequality shows that(82)E42+E43≲lnnpsupj0≤j≤j12jp/2−1∑k=02j−1Eb∧j,k−bj,k2p1/2EIk∈Dj1/2.
Note thatEIk∈Dj=ℙb^j,k−bj,k≥λj/2. Then, we obtain
(83)E42+E43≲lnnpsupj0≤j≤j12jp/2−12j2−ρ2j/41/22jvℓ∗+dpn−p/2≲lnnpsupj0≤j≤j1n−p/22jp/2+vℓ∗p+dp−ρ2/8,
by Lemma4 and Lemma 8.
Choosingρ large enough such that p/2+vℓ∗p+dp≤ρ2/8, then
(84)E42+E43≲lnnpn−p/22j0p/2+vℓ∗p+dp−ρ2/8≲lnnpn−p/22pj0vℓ∗+d+1/2.
Similar to previous discussions in (71)–(73), we conclude that
(85)E42+E43≲lnnpn−sp/2s+2vℓ∗+2d+1,s≥2vℓ∗+2d+12pr−1,lnnpn−s′p/2s−1/r+2vℓ∗+2d+1,s<2vℓ∗+2d+12pr−1.
The main part that remains is estimating termE41+E44.
(1)
Ifs<vℓ∗+d+1/2p/r−1, namely, r<2vℓ∗+2d+1p/2s+2vℓ∗+2d+1; then for any 1≤r<p~<p, we have Ibj,k≥λj/2≤2bj,k/λjp~ and Ibj,k<2λj≤2λj/bj,kp−p~. By (27), (28), and Lemma 2 (ii), one has λj=ρjVarb^j,k. Therefore, we get(86)∑k=02j−1Eb∧j,k−bj,kpIk∈Bj≲∑k=02j−1Varb∧j,kp/2bj,kλj/2p~≲∑k=02j−1Varb∧j,kp−p~/2bj,kp~≲n−p−p~/22jvℓ∗+dp−p~∑k=02j−1bj,kp~.
Meanwhile,(87)∑k=02j−1bj,kpIk∈Cjc≲∑k=02j−1bj,kp2λjbj,kp−p~≲∑k=02j−1bj,kp~Varb∧j,kjp−p~/2≲jp−p~/2n−p−p~/22jvℓ∗+dp−p~∑k=02j−1bj,kp~.
Note that∑k=02j−1bj,kp~1/p~≲∑k=02j−1bj,kr1/r for r<p~. Then it follows from (86) and (87) that
(88)E41+E44≲lnn3p−p~/2n−p−p~/2supj0≤j≤j12jp/2−12jvℓ∗+dp−p~∑k=02j−1bj,kp~≲lnn3p/2n−p−p~/2supj0≤j≤j12jp−2+2vℓ∗p+2dp/2p~2−jvℓ∗+d∑k=02j−1bj,kr1/rp~.
Takingp~=p+2vℓ∗p+2dp−2/2s−1/r+2vℓ∗+2d+1, one obtains 2jp−2+2vℓ∗p+2dp/2p~2−jvℓ∗+d=2js−1/r+1/2 and
(89)2js−1/r+1/2∑k=02j−1bj,kr1/r≤fdBr,qs.
Furthermore,p−p~/2=s′p/2s−1/r+2vℓ∗+2d+1. Therefore,
(90)E41+E44≲lnn3p/2n−s′p/2s−1/r+2vℓ∗+2d+1.(2)
Ifs≥vℓ∗+d+1/2p/r−1, then j0≤j0∗≤j1 with 2j0∗ in (72).
Case 1: For j0≤j≤j0∗, due to Lemma 4, we know that
(91)E411≔lnnpsupj0≤j≤j0∗2jp/2−1∑k=02j−1Eb∧j,k−bj,kpIk∈Aj∩Bj≲lnnpsupj0≤j≤j0∗2jp/2−1∑k=02j−1Eb∧j,k−bj,kp≲lnnpsupj0≤j≤j0∗2jp/2−12j2pjvℓ∗+dn−p/2≲lnnpn−p/22pj0∗vℓ∗+d+1/2.
Meanwhile,∑k=02j−1bj,kpIk∈Ajc∩Cjc≲∑k∈Cjc2λjp for bj,k<2λj. Hence,
(92)E441≔lnnpsupj0≤j≤j0∗2jp/2−1∑k=02j−1Ebj,kpIk∈Ajc∩Cjc≲lnnpsupj0≤j≤j0∗jp/22jp/2−12j2pjvℓ∗+dn−p/2≲lnn3p/2n−p/22pj0∗vℓ∗+d+1/2.
According to (72), we obtain
(93)E411+E441≲lnn3p/2n−sp/2s+2vℓ∗+2d+1.
Case 2: For j0∗≤j≤j1, by bj,k≥λj/2 and Lemma 4, we have
(94)E412≔lnnpsupj0∗≤j≤j12jp/2−1∑k=02j−1Eb∧j,k−bj,kpIk∈Aj∩Bj≲lnnpsupj0∗≤j≤j12jp/2−1∑k=02j−1Varb∧j,kp/22bj,kλjr≲lnnpsupj0∗≤j≤j12jp/2−1∑k=02j−1Varb∧j,kp−r/2bj,kr≲lnnpn−p−r/2supj0∗≤j≤j12jp/2−12p−rvℓ∗+dj∑k=02j−1bj,kr.
On the other hand,(95)∑k=02j−1bj,kpIk∈Ajc∩Cjc≤∑k∈Cjcbj,kp−rbj,kr≤2λjp−r∑k=02j−1bj,kr≲lnnp−r/2n−p−r/22p−rvℓ∗+dj∑k=02j−1bj,kr.
Hence,(96)E442≔lnnpsupj0∗≤j≤j12jp/2−1∑k=02j−1Ebj,kpIk∈Ajc∩Cjc≲lnn3p/2n−p−r/2supj0∗≤j≤j12jp/2−12p−rvℓ∗+dj∑k=02j−1bj,kr.
Since(97)2js+j1/2−1/r∑k=02j−1bj,kr1/r≤fdBr,qs, we obtain
(98)E412+E442≲lnn3p/2n−p−r/2supj0∗≤j≤j12−jN, where N=sr−p−rvℓ∗+d+1/2. Note that N≥0 follows from s≥vℓ∗+d+1/2p/r−1. Then
(99)E412+E442≲lnn3p/2n−p−r/22−j0∗N.
Taking2j0∗ as in (72), we find that
(100)E412+E442≲lnn3p/2n−p−r/22−j0∗N≲lnn3p/2n−sp/2s+2vℓ∗+2d+1.
This with (93) shows
(101)E41+E44≤E411+E441+E412+E442≲lnn3p/2n−sp/2s+2vℓ∗+2d+1.Remark 10.
Theorem9 shows that the convergence rate of nonlinear estimator changes as s decreases. Now, we discuss the convergence rates in the regular case from Theorem 6 (i) and Theorem 9. For r≥p, the nonlinear estimator has the same convergence rate as the linear one up to a logarithmic factor; for 1≤r<p, the nonlinear estimator performs better than the linear one. Moreover, the nonlinear wavelet estimator is adaptive.


## 6. Conclusions

In this paper, we consider the wavelet estimation of function derivatives from a multichannel deconvolution model. We construct the linear wavelet estimator by wavelet projection method and prove its upper bounds ofLp- 1≤p<∞ risk. It turns out that the regular-smooth case has a better convergence rate than the super-smooth one. Note that the linear estimator is only adaptive in the super-smooth case. Then, we construct the adaptive nonlinear wavelet estimator by hard-thresholding method and provide its upper bounds of Lp- 1≤p<∞ risk. If d=0, our convergence rates of the nonlinear estimator agree with results of [9] up to a logarithmic factor. In addition, our result generalizes Theorem 9 of [17] from L2- to Lp-risk case up to a logarithmic factor when M=1.We consider both regular-smooth and super-smooth cases, while [18, 20] focus on the regular-smooth one. They mainly consider the nonlinear wavelet estimator. We also provide the linear wavelet estimator, which is simple and easy to compute. We want to point out that this paper is a theoretical exploration. It is interesting to give numerical experiments or some applications. We will investigate these aspects in future.



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*Source: 2075229-2022-12-31.xml*