Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises

Abstract

Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a density 𝑓 based on an independent and identically distributed sample 𝑌₁,…,𝑌_𝑛 from 𝑔=𝑓∗𝜑. We show a wavelet optimal estimation for a density (function) over Besov ball 𝐵^𝑠_𝑟,𝑞(𝐿) and 𝐿^𝑝 risk (1≤𝑝<∞) in the presence of severely ill-posed noises. A wavelet linear estimation is firstly presented. Then, we prove a lower bound, which shows our wavelet estimator optimal. In other words, nonlinear wavelet estimations are not needed in that case. It turns out that our results extend some theorems of Pensky and Vidakovic (1999), as well as Fan and Koo (2002).

1. Introduction and Preliminary

Wavelets have made great achievements in studying the statistical model 𝑌=𝑋+𝜖, where 𝑋 stands for real-valued random variable with unknown probability density 𝑓, and 𝜖 denotes an independent random noise (error) with density 𝜑.

In 1999, Pensky and Vidakovic [1] investigate Meyer wavelet estimation over Sobolev spaces 𝑊^𝑠₂(ℝ) and 𝐿² risk under moderately and severely ill-posed noises. Three years later, Fan and Koo [2] extend those works from 𝑊^𝑠₂(ℝ) to Besov spaces 𝐵^𝑠_𝑟,𝑞(ℝ)(1≤𝑟≤2). It should be pointed out that, by using different method, Lounici and Nickl [3] study wavelet optimal estimation over Besov spaces 𝐵^𝑠_∞,∞(ℝ) and 𝐿^∞ risk under both noises. In [4], we provide a wavelet optimal estimation over 𝐵^𝑠_𝑟,𝑞(ℝ) and 𝐿^𝑝 risk (1≤𝑝<∞,𝑟,𝑞∈[1,∞]) under moderately ill-posed noise. This current paper deals with the same problem under the severely ill-posed noises. It turns out that our result contains some theorems of [1, 2] as special cases. Our discussion also shows that nonlinear wavelet estimations are not needed for severely ill-posed noise, which is totally different with moderately ill-posed case.

Let 𝜙 and 𝜓∈𝐿²(ℝ) be a scaling and mother wavelet function, respectively. Then each 𝑓∈𝐿²(ℝ) has an expansion (in 𝐿² sense): 𝑓=_𝑘∈ℤ𝛼_𝐽𝑘𝜙_𝐽𝑘+_{𝑗≥𝐽𝑘∈ℤ}𝛽_𝑗𝑘𝜓_𝑗𝑘, (1) with 𝛼_𝑗𝑘∶=⟨𝑓,𝜙_𝑗𝑘⟩ and 𝛽_𝑗𝑘∶=⟨𝑓,𝜓_𝑗𝑘⟩. Here and throughout, we use the standard notation ℎ_𝑗𝑘(𝑥)∶=2^𝑗/2ℎ(2^𝑗𝑥−𝑘) in wavelet analysis [5]. A class of important wavelets are Meyer's, whose Fourier transforms are 𝐶^∞ and compactly supported on {𝑡∶2𝜋/3≤|𝑡|≤8𝜋/3} [5]. It is easy to see that ∀𝑎≥0, ∃𝐶_𝑎>0 such that |𝑥|^𝑎|𝜙(𝑥)|≤𝐶_𝑎. In this paper, the Fourier transform 𝑓 for 𝑓∈𝐿(ℝ) is defined by 𝑓(𝑡)∶=𝐹𝑓(𝑡)=_ℝ𝑓(𝑥)𝑒^{−𝑖𝑡𝑥}𝑑𝑥. (2) The classical method extends that definition to 𝐿²(ℝ) functions.

The following two lemmas are fundamental in our discussions. We use ‖𝑓‖_𝑝 to denote 𝐿^𝑝(ℝ) norm of 𝑓∈𝐿^𝑝(ℝ), and ‖𝜆‖_𝑝 do 𝑙^𝑝(ℤ) norm of 𝜆∈𝑙^𝑝(ℤ), where 𝑙^𝑝(ℤ)𝜆∶=𝜆=_𝑘,∑_𝑘∈ℤ𝜆_𝑘^𝑝𝜆<∞,1≤𝑝<∞,𝜆=_𝑘,sup_𝑘∈ℤ𝜆_𝑘<∞,𝑝=∞. (3)

Lemma 1 (see [6]). Let ℎ be a scaling or a wavelet function with sup_𝑥∈ℝ∑_𝑘∈ℤ|ℎ(𝑥−𝑘)|<∞. Then, there exist 𝐶₂≥𝐶₁>0 such that for 𝜆={𝜆_𝑘}∈𝑙^𝑝(ℤ) with 1≤𝑝≤∞, 𝐶₁2^{𝑗(1/2−1/𝑝)}‖𝜆‖_𝑝≤_𝑘∈ℤ𝜆_𝑘ℎ_𝑗𝑘𝑝≤𝐶₂2^{𝑗(1/2−1/𝑝)}‖𝜆‖_𝑝. (4)

One of the advantages of wavelet bases is that they can characterize Besov spaces. To introduce those spaces [6], we need the well-known Sobolev spaces with integer exponents 𝑊^𝑚_𝑝(ℝ)∶=𝑓∈𝐿^𝑝(ℝ),𝑓^(𝑚)∈𝐿^𝑝(ℝ), (5) with the Sobolev norm ‖𝑓‖_𝑊𝑚𝑝∶=‖𝑓‖_𝑝+‖𝑓^(𝑚)‖_𝑝. Then 𝐿^𝑝(ℝ) can be considered as 𝑊_𝑝⁽⁰⁾(ℝ). For 1≤𝑝, 𝑞≤∞ and 𝑠=𝑚+𝛼 with 𝛼∈(0,1], a Besov space is defined by 𝐵^𝑠_𝑟,𝑞(ℝ)∶=𝑓∈𝑊^𝑚_𝑟(ℝ),𝑡^−𝛼𝜔²_𝑟𝑓^(𝑚),𝑡^∗_𝑞<∞, (6) with the norm ‖𝑓‖_𝑠𝑟𝑞∶=‖𝑓‖_𝑊𝑚𝑟+‖𝑡^−𝛼𝜔²_𝑟(𝑓^(𝑚),𝑡)‖^∗_𝑞, where 𝜔²_𝑟(𝑓,𝑡)∶=sup_|ℎ|≤𝑡‖𝑓(⋅+2ℎ)−2𝑓(⋅+ℎ)+𝑓(⋅)‖_𝑟 denotes the smoothness modulus of 𝑓 and ‖ℎ‖^∗_𝑞=∫^∞₀|ℎ(𝑡)|^𝑞𝑡𝑑𝑡^1/𝑞,if1≤𝑞<∞,esssup_𝑡∈ℝ|ℎ(𝑡)|,if𝑞=∞. (7)

Lemma 2 (see [6]). Let 𝜙 be a Meyer scaling function and 𝜓 be the corresponding wavelet. If 𝑓∈𝐿^𝑟(ℝ), 1≤𝑟≤∞, 𝛼_0𝑘=∫𝑓(𝑥)𝜙_0𝑘(𝑥)𝑑𝑥, and 𝛽_𝑗𝑘=∫𝑓(𝑥)𝜓_𝑗𝑘(𝑥)𝑑𝑥, then the following assertions are equivalent:(i)𝑓∈𝐵^𝑠_𝑟,𝑞(ℝ),1≤𝑞≤∞;(ii){2^𝑗𝑠‖𝑃_𝑗𝑓−𝑓‖_𝑟}_𝑗≥0∈𝑙^𝑞, where 𝑃_𝑗𝑓∶=∑_𝑘∈ℤ𝛼_𝑗𝑘𝜙_𝑗𝑘;(iii)‖𝛼_0𝑘‖_𝑟+‖{2^{𝑗(𝑠+1/2−1/𝑟)}‖𝛽_𝑗⋅‖_𝑟}_𝑗≥0‖_𝑞<∞.

In each case, 𝑓_𝑠𝑟𝑞∼𝛼_0⋅𝑟+{2^{𝑗(𝑠+1/2−1/𝑟)}𝛽_𝑗⋅𝑟}_𝑗≥0𝑞. (8) Here and after, 𝐴≲𝐵 denotes 𝐴≤𝐶𝐵 for some constant 𝐶>0; 𝐴≳𝐵 means 𝐵≲𝐴; 𝐴∼𝐵 stands for both 𝐴≲𝐵 and 𝐵≲𝐴, 𝛼_0⋅ does for the sequence of {𝛼_0𝑘,𝑘∈ℤ}.

At the end of this subsection, we make some assumptions on noise density 𝜑, which will be dealt with in this current paper. For 𝛼>0, 𝑐>0, 𝛽∈ℝ,(C1)|𝜑(𝑡)|≤𝐶(1+|𝑡|²)^−𝛽/2𝑒^{−𝑐|𝑡|𝛼}(a.e.); (C2)|𝜑^′(𝑡)|≤𝐶(1+|𝑡|²)^−𝛽/2𝑒^{−𝑐|𝑡|𝛼}(a.e.); (C3)|𝜑(𝑡)|≥𝐶(1+|𝑡|²)^−𝛽/2𝑒^{−𝑐|𝑡|𝛼}(a.e.). Clearly, the classical Cauchy densities satisfy (C1)–(C3) with 𝛼=𝑐=1 and 𝛽=0, and the Gaussian density does satisfy (C1)–(C3) with 𝛼=2, 𝑐=1/2, and 𝛽=0. It should be pointed out that those above conditions (C1)–(C3) are a little different with [2].

In the next section, we define a wavelet linear estimator and provide an upper bound estimation over Besov spaces 𝐵^𝑠_𝑟,𝑞(ℝ) and 𝐿^𝑝 risk under the condition (C3); the third part gives a lower bound estimation which shows the result of Section 2 optimal; some concluding remarks are discussed in the last part.

2. Upper Bound

To introduce the main theorem of this section, we assume that 𝑌₁,𝑌₂,…,𝑌_𝑛 are independent and identically distributed (i.i.d) random variables of 𝑌=𝑋+𝜖, the density 𝜑 of random noise 𝜖 satisfies condition (C3), and 𝜙 stands for the Meyer scaling function. As in [1], define 𝐊_𝑗𝜙𝑦1∶=2𝜋_ℝ𝑒^𝑖𝑡𝑦𝜙(𝑡)𝜑−2^𝑗𝑡𝛼𝑑𝑡,_𝑗𝑘1∶=𝑛^𝑛_𝑖=1𝐊_𝑗𝜙_𝑗𝑘𝑌_𝑖, (9) as well as a linear wavelet estimator 𝑓^𝐿_𝑛∶=_{|𝑘|≤𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘, (10) (the positive integer 𝐾_𝑛 will be given later on). Then 𝛼_𝑗⋅∈𝑙^∞, 𝑓^𝐿_𝑛 is well defined, and 𝐸𝑓^𝐿_𝑛=∑_{|𝑘|≤𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘.

We use supp 𝑓 to stand for the support of 𝑓 and |supp𝑓| to do its length. Moreover, for 𝐿>0, denote 𝐵^𝑠_𝑟,𝑞(𝐿)∶=𝑓∈𝐵^𝑠_𝑟,𝑞(ℝ),𝑓≥0,_ℝ𝑓(𝑥)𝑑𝑥=1and𝑓_𝑠𝑟𝑞,𝐵≤𝐿_{𝑠𝑟,𝑞}(𝐿,𝑀)=𝑓∈𝐵^𝑠_𝑟,𝑞(𝐿),supp𝑓≤𝑀forsome𝑀>0. (11) It is reasonable to assume 𝐿>1 for 𝑟=1, since ‖𝑓‖_𝑠1𝑞≥‖𝑓‖₁=1 in that case.

Theorem 3. Let 𝜑 satisfy (C3) and 𝜙 be the Meyer scaling function. If 𝑝∈[1,∞), 𝑞,𝑟∈[1,∞], ‖𝑥²𝑓‖_𝜇≤𝐴(𝜇≥1,𝐴>0), then, with 𝐾_𝑛∼𝑒^(ln𝑛)𝜃(0<𝜃<1), (3/8𝜋)((ln𝑛)/4𝑐)^1/𝛼 < 2^𝑗≤(3/4𝜋)((ln𝑛)/4𝑐)^1/𝛼, 𝑠^′∶=𝑠−(1/𝑟−1/𝑝)₊, and 𝑥₊=max{𝑥,0}, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿,𝑀)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲(ln𝑛)^{−(𝑠′/𝛼)𝑝}. (12) In particular, 𝑓∈𝐵^𝑠_𝑟,𝑞(𝐿,𝑀) can be replaced by 𝑓∈𝐵^𝑠_𝑟,𝑞(𝐿), when 𝑟≤𝑝.

Proof. When 𝑟≤𝑝, 𝑠^′∶=𝑠−(1/𝑟−1/𝑝)₊=𝑠−1/𝑟+1/𝑝. Since 𝑙^𝑟 is continuously embedded to 𝑙^𝑝, Lemma 2 implies 𝐵^𝑠_𝑟,𝑞(ℝ)⊆𝐵^𝑠′_𝑝,𝑞(ℝ). Hence, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲sup_{𝑓∈𝐵𝑠′𝑝,𝑞(𝐿)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝. (13) When 𝑟>𝑝, one obtains that, for some 𝐶>0, 𝐵^𝑠_𝑟,𝑞(𝐿,𝑀)⊆𝐵^𝑠_𝑝,𝑞(𝐶𝐿,𝑀) and sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿,𝑀)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲sup_{𝑓∈𝐵𝑠𝑝,𝑞(𝐶𝐿,𝑀)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝. (14) In fact, 𝑓∈𝐵^𝑠_𝑟,𝑞(𝐿,𝑀) and Hölder inequality imply that ‖𝑓‖^𝑝_𝑝=∫|𝑓(𝑥)|^𝑝𝑑𝑥≲‖𝑓(𝑥)‖^𝑝_𝑟 due to |supp𝑓|≤𝑀. By the definition of Besov norm, ‖𝑓‖_𝑠𝑝𝑞≤𝐶‖𝑓‖_𝑠𝑟𝑞. According to (13) and (14), it is sufficient to prove sup_{𝑓∈𝐵𝑠𝑝,𝑞(𝐿)}𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲(ln𝑛)^{−(𝑠/𝛼)𝑝}, (15) for the conclusion of Theorem 3.
Recall that 𝑓^𝐿_𝑛=∑_{|𝑘|≤𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘 and 𝐸𝑓^𝐿_𝑛=∑_{|𝑘|≤𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘. Then 𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲𝐸𝑓^𝐿_𝑛−𝐸𝑓^𝐿_𝑛^𝑝_𝑝+_{|𝑘|>𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘^𝑝_𝑝+𝑃_𝑗𝑓−𝑓^𝑝_𝑝. (16) By 𝑓∈𝐵^𝑠_𝑝,𝑞(𝐿) and Lemma 2, 𝑃_𝑗𝑓−𝑓_𝑝≲2^−𝑗𝑠. (17) To estimate the middle term of (16), one observes that 𝛼_𝑗𝑘=∫_ℝ𝜙_𝑗𝑘(𝑥)𝑓(𝑥)𝑑𝑥, |𝑘²𝛼_𝑗𝑘| ≤ ∫_ℝ|2^𝑗𝑥−𝑘−2^𝑗𝑥|²|𝜙_𝑗𝑘(𝑥)|𝑓(𝑥)𝑑𝑥≲∫_ℝ|2^𝑗𝑥−𝑘|²|𝜙_𝑗𝑘(𝑥)|𝑓(𝑥)𝑑𝑥+∫_ℝ|2^𝑗𝑥|²|𝜙_𝑗𝑘(𝑥)|𝑓(𝑥)𝑑𝑥. Since 𝜙 is the Meyer scaling function, sup_𝑥∈ℝ|𝑥|²|𝜙(𝑥)|<∞ and _ℝ2^𝑗𝑥−𝑘²𝜙_𝑗𝑘(𝑥)𝑓(𝑥)𝑑𝑥≤2^𝑗/2sup_𝑥∈ℝ|𝑥|²𝜙(𝑥). (18) On the other hand, ∫_ℝ|2^𝑗𝑥|²|𝜙_𝑗𝑘(𝑥)|𝑓(𝑥)𝑑𝑥≤2^2𝑗‖𝑥²𝑓‖_𝜇×‖𝜙_𝑗𝑘‖_𝜇′≲2^{𝑗(5/2−1/𝜇′)}‖𝑥²𝑓‖_𝜇 with 1/𝜇 + 1/𝜇^′=1. Therefore |𝑘²𝛼_𝑗𝑘|≲2^𝑗/2+2^{𝑗(5/2−1/𝜇′)}≲2^𝑗(5/2) and ∑_{|𝑘|>𝐾𝑛}|𝛼_𝑗𝑘|^𝑝 ≲ ∑_{|𝑘|>𝐾𝑛}𝑘^−2𝑝2^{𝑗(5/2)𝑝} ≲ 𝐾_𝑛^1−2𝑝2^{𝑗(5/2)𝑝}. This with Lemma 1 leads to _{|𝑘|>𝐾𝑛}𝛼_𝑗𝑘𝜙_𝑗𝑘^𝑝_𝑝≲2^{𝑗(𝑝/2−1)}_{|𝑘|>𝐾𝑛}𝛼_𝑗𝑘^𝑝≤2^{𝑗(3𝑝−1)}𝐾_𝑛^1−2𝑝. (19)
Now, it remains to consider 𝐸‖𝑓^𝐿_𝑛−𝐸𝑓^𝐿_𝑛‖^𝑝_𝑝: Using 𝑓^𝐿_𝑛−𝐸𝑓^𝐿_𝑛=∑_{|𝑘|≤𝐾𝑛}(𝛼_𝑗𝑘−𝛼_𝑗𝑘)𝜙_𝑗𝑘 and Lemma 1, one knows 𝐸𝑓^𝐿_𝑛−𝐸𝑓^𝐿_𝑛^𝑝_𝑝≲2^{𝑗(𝑝/2−1)}_{|𝑘|≤𝐾𝑛}𝐸𝛼_𝑗𝑘−𝛼_𝑗𝑘^𝑝. (20) Clearly, 𝛼_𝑗𝑘−𝛼_𝑗𝑘=(1/𝑛)∑^𝑛_𝑖=1[(𝐊_𝑗𝜙)_𝑗𝑘(𝑌_𝑖)−𝐸(𝐊_𝑗𝜙)_𝑗𝑘(𝑌_𝑖)]. Define 𝑋_𝑖,𝑘∶=(𝐊_𝑗𝜙)_𝑗𝑘(𝑌_𝑖)−𝐸(𝐊_𝑗𝜙)_𝑗𝑘(𝑌_𝑖). Then 𝐸𝑋_𝑖,𝑘=0 and 𝛼_𝑗𝑘−𝛼_𝑗𝑘=(1/𝑛)∑^𝑛_𝑖=1𝑋_𝑖,𝑘. To apply Rosenthal's inequality (Proposition 10.2, [6]), one estimates |𝑋_𝑖,𝑘| and 𝐸|𝑋_𝑖,𝑘|^𝑝: note that |(𝐊_𝑗𝜙)_𝑗𝑘(𝑌_𝑖)|=2^𝑗/2|(1/2𝜋) × ∫_ℝ𝑒^{𝑖𝑡(2𝑗𝑌𝑖−𝑘)}(𝜙(𝑡)/𝜑(−2^𝑗𝑡))𝑑𝑡|≲2^𝑗/2∫_ℝ(1+|2^𝑗𝑡|²)^𝛽/2𝑒^{𝑐|2𝑗𝑡|𝛼}×|𝜙(𝑡)|𝑑𝑡≲2^{𝑗(𝛽+1/2)}𝑒^{𝑐((4𝜋/3)2𝑗)𝛼} due to (C3) and supp𝜙⊆[−(4/3)𝜋,(4/3)𝜋]. Then 𝑋_𝑖,𝑘≲2^{𝑗(𝛽+1/2)}𝑒^{𝑐((4𝜋/3)2𝑗)𝛼},𝐸𝑋_𝑖,𝑘^𝑝≲2^{𝑗(𝛽+1/2)𝑝}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}. (21)
Because 𝑋_𝑖,𝑘 are i.i.d, the Rosenthal's inequality tells that 𝐸𝛼_𝑗𝑘−𝛼_𝑗𝑘^𝑝≲𝑛^−𝑝𝑛𝐸𝑋_𝑖,𝑘^𝑝+𝑛^𝑝/2𝐸𝑋_𝑖,𝑘^2𝑝/2𝑛,𝑝>2,_−𝑝/2𝐸𝑋_𝑖,𝑘^2𝑝/2,0<𝑝≤2. (22) This with (21) implies that, for 𝑝≥2, 𝐸|𝛼_𝑗𝑘−𝛼_𝑗𝑘|^𝑝≲𝑛^1−𝑝2^{𝑗(𝛽+1/2)𝑝}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}𝐼{𝑝 > 2} + 𝑛^−𝑝/22^{𝑗(𝛽+1/2)𝑝}×𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}≲𝑛^−𝑝/22^{𝑗(𝛽+1/2)𝑝}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}. Moveover, (20) reduces to 𝐸𝑓^𝐿_𝑛−𝐸𝑓^𝐿_𝑛^𝑝_𝑝≲𝐾_𝑛𝑛^−𝑝/22^{𝑗(𝛽+1)𝑝−𝑗}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}. (23) Then it follows from (16)–(19) and (23) that 𝐸𝑓^𝐿_𝑛−𝑓^𝑝_𝑝≲2^{−𝑗𝑝𝑠}+2^{𝑗(3𝑝−1)}𝐾_𝑛^1−2𝑝+𝐾_𝑛𝑛^−𝑝/22^{𝑗(𝛽+1)𝑝−𝑗}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}. (24) By the choices 𝐾_𝑛∼𝑒^(ln𝑛)𝜃 and (3/8𝜋)((ln𝑛)/4𝑐)^1/𝛼<2^𝑗≤(3/4𝜋)((ln𝑛)/4𝑐)^1/𝛼 (stated in Theorem 3), one receives that 2^{−𝑗𝑝𝑠}≲(ln𝑛)^{−(𝑠/𝛼)𝑝}, 2^{𝑗(3𝑝−1)}𝐾_𝑛^1−2𝑝≲(ln𝑛)^{(3𝑝−1)/𝛼}𝑒^{−(2𝑝−1)(ln𝑛)𝜃}=𝑜(ln𝑛)^{−(𝑠/𝛼)𝑝},𝐾_𝑛𝑛^−𝑝/22^{𝑗(𝛽+1)𝑝−𝑗}𝑒^{𝑐𝑝((4𝜋/3)2𝑗)𝛼}≲𝑒^(ln𝑛)𝜃𝑛^−𝑝/2(ln𝑛)^{((𝛽+1)𝑝−1)/𝛼}𝑛^𝑝/4=𝑜(ln𝑛)^{−(𝑠/𝛼)𝑝}. (25) Finally, the desired conclusion (15) follows.

Remark 4. Note that the choices of 𝑗 and 𝐾_𝑛 do not depend on the unknown parameters 𝑠, 𝑟, and 𝑞. Then our linear wavelet estimator 𝑓^𝐿_𝑛 over Besov space 𝐵^𝑠_𝑟,𝑞 is adaptive or implementable. The same conclusion holds for 𝐿^∞ and 𝐿² estimations; see Theorem 2 in [3] and Corollary 1 in [1]. On the other hand, when 𝑝=2 and 1≤𝑟≤2, our Theorem 3 reduces to Theorem 4 in [2]; from the proof of Theorem 3, we find that, for 𝑝>1, the assumption ‖𝑥²𝑓(𝑥)‖_𝜇≤𝐴 can be replaced by ‖𝑥𝑓(𝑥)‖_∞≤𝐴, which is the same as in [1]. Therefore, for 𝑝=𝑟=𝑞=2, Theorem 3 of [1] follows directly from our Theorem 3.

3. Lower Bound

In this part, we will provide a lower bound estimation, which shows Theorem 3 to be the best possible in some sense. The following lemmas are needed in the proof of our main theorem of this section.

Lemma 5. Let ℎ_𝜂(𝑥)∶=𝜂𝑝(𝜂𝑥) with 𝑝(𝑥)=1/𝜋(1+𝑥²), 𝜂>0, and 𝑟,𝑞≥1. Then for 𝐿>0 (𝐿>2 when 𝑟=1), there exists 𝜂₀>0 such that ℎ₀∶=ℎ_𝜂0∈𝐵^𝑠_𝑟,𝑞(𝐿/2). If 𝜓 is the Meyer wavelet function and |𝜆_𝑘|≤𝑑2^−𝑗/2(𝑘=1,2,…,2^𝑗), then, for some small 𝑑>0, ^2𝑗_𝑘=1𝜆_𝑘𝜓_𝑗𝑘(𝑥)≤ℎ₀(𝑥). (26)

Proof. It is easy to see that (1+𝑥²)⁻¹∈𝑊^𝑚_𝑟(ℝ) (for 𝑟≥1) and ℎ_𝜂∈𝐵^𝑠_𝑟,𝑞(ℝ)(𝑠<𝑚) by the definition of Besov space. Since ℎ_{𝜂𝑠𝑟𝑞}=𝜂^1−1/𝑟𝑝_𝑟+𝜂^{1+⌊𝑠⌋−1/𝑟}𝑝^(⌊𝑠⌋)_𝑟+𝜂^{1+𝑠−1/𝑟}𝜔²_𝑟(𝑝^(⌊𝑠⌋),𝑡)𝑡^{𝑠−⌊𝑠⌋∗}_𝑞, (27) where ⌊𝑠⌋ denotes the largest integer no more than 𝑠,‖ℎ₀‖_𝑠𝑟𝑞 can be made small enough by choosing small 𝜂₀>0, when 𝑟>1. Clearly, 𝐿>2 is needed, when 𝑟=1.
If |𝜆_𝑘|≤𝑑2^−𝑗/2(𝑘=1,2,…,2^𝑗), then |∑^2𝑗_𝑘=1𝜆_𝑘𝜓_𝑗𝑘(𝑥)|≤𝑑∑^2𝑗_𝑘=1|𝜓(2^𝑗𝑥−𝑘)|≲𝑑∑^2𝑗_𝑘=1(1+|2^𝑗𝑥−𝑘|²)⁻¹ because 𝜓 is the Meyer function. Note that ∑^2𝑗_𝑘=1(1+|2^𝑗𝑥−𝑘|²)⁻¹=∑^2𝑗_𝑘=1(1+2^2𝑗|𝑥 − 2^−𝑗𝑘|²)⁻¹ ≤ 2^−2𝑗∑^2𝑗_𝑘=1(|𝑥|²−1)⁻¹=2^−𝑗(𝑥²−1)⁻¹. Then for some small 𝑑>0 and |𝑥|≥2, ^2𝑗_𝑘=1𝜆_𝑘𝜓_𝑗𝑘(𝑥)≤2^−𝑗ℎ₀(𝑥). (28) Hence, (26) holds for |𝑥|≥2. On the other hand, when |𝑥|<2, ℎ₀(𝑥)≳1 and |∑^2𝑗_𝑘=1𝜆_𝑘𝜓_𝑗𝑘(𝑥)| ≤ 𝑑∑^2𝑗_𝑘=1|𝜓(2^𝑗𝑥−𝑘)|≤𝑑sup_𝑥∑_𝑘∈ℤ|𝜓(𝑥−𝑘)|≤𝐶𝑑≤ℎ₀(𝑥). Therefore, (26) is true, when 𝑑>0 small enough. This completes the proof of Lemma 5.

The next lemma extends an estimate in the proof of Theorem 1 in [3].

Lemma 6. Let 𝜓 be the Meyer wavelet function, ℎ₀(𝑥), defined as in Lemma 5. If 𝜑 satisfies (C1), (C2), and 𝜔_𝑘∈{0,1}, then _ℝℎ₀∗𝜑⁻¹𝑦^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑²𝑦𝑑𝑦≲2^−2𝑗𝛽𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼2𝑗}_𝑘=1𝜔²_𝑘. (29)

Proof. As shown in proof of Theorem 1 of [3], one finds easily that (ℎ₀∗𝜑)(𝑦)≳(1+𝑦²)⁻¹ and therefore _ℝℎ₀∗𝜑⁻¹𝑦^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑²𝑦≲𝑑𝑦_ℝ1+𝑦^22𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑²𝑦𝑑𝑦. (30) By Parseval identity, (C1) and supp𝜓_𝑗𝑘⊆{𝑦,|𝑦|≥(2𝜋/3)2^𝑗}, ∫_ℝ(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)²(𝑦)𝑑𝑦 = (1/2𝜋)∫_{|𝑡|≥(2𝜋/3)2𝑗}×|∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘(𝑡)|²|𝜑(𝑡)|²𝑑𝑡≲2^−2𝛽𝑗𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼}∫_{|𝑡|≥(2𝜋/3)2𝑗}×|∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘(𝑡)|²𝑑𝑡. Moreover, the orthonormality of 𝜓_𝑗𝑘 concludes that _ℝ^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑²𝑦𝑑𝑦≲𝐶2^−2𝛽𝑗𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼2𝑗}_𝑘=1𝜔²_𝑘. (31)
To estimate ∫_ℝ𝑦²(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)²(𝑦)𝑑𝑦, one proves an inequality: 𝐼∶=_ℝ^2𝑗_𝑘=1𝜔_𝑘𝑑𝜓𝑑𝑡^𝑗𝑘(𝑡)²𝑑𝑡≤𝐶^2𝑗_𝑘=1𝜔²_𝑘. (32) Note that ∑^2𝑗_𝑘=1𝜔_𝑘𝜓_0𝑘(𝑥),  𝑥∑^2𝑗_𝑘=1𝜔_𝑘𝜓_0𝑘(𝑥)∈𝐿(ℝ), and ∑^2𝑗_𝑘=1𝜔_𝑘(𝑑/𝑑𝑡)𝜓_𝑗𝑘(𝑡) = 2^−(3/2)𝑗[(𝑑/𝑑𝑡)𝐹(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_0𝑘)](2^−𝑗𝑡). Then ∫_ℝ[(𝑑/𝑑𝑡)𝐹(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_0𝑘)]²(𝑡)𝑑𝑡=∫_ℝ|𝑖𝑥|²|∑^2𝑗_𝑘=1𝜔_𝑘 × 𝜓_0𝑘(𝑥)|²𝑑𝑥=∑^2𝑗_𝑘=1∑^2𝑗_𝑘′=1𝜔_𝑘𝜔_𝑘′∫_ℝ(𝑥 + 𝑘)²𝜓(𝑥)𝜓[𝑥−(𝑘^′−𝑘)]𝑑𝑥 and 𝐼=2^{2−2𝑗𝑗2}_𝑘=1^𝑗_𝑘′=1𝜔_𝑘𝜔_𝑘′ℝ(𝑥+𝑘)²𝜓(𝑥)𝜓𝑥−𝑘^′−𝑘𝑑𝑥. (33) Since ⟨𝜓(𝑥),𝜓(𝑥−𝑘)⟩=𝛿_𝑘,0, ∑^2𝑗_𝑘=1∑^2𝑗_𝑘′=1𝜔_𝑘𝜔_𝑘′𝑘²∫_ℝ𝜓(𝑥)𝜓(𝑥+𝑘−𝑘^′)𝑑𝑥=∑^2𝑗_𝑘=1𝜔²_𝑘𝑘²≤2^2𝑗∑^2𝑗_𝑘=1𝜔²_𝑘; On the other hand, the boundedness of ∫_ℝ𝑥²𝜓(𝑥)𝜓(𝑥+𝑙)𝑑𝑥 and ∫_ℝ𝑥𝜓(𝑥)𝜓(𝑥+𝑙)𝑑𝑥 implies that ^2𝑗2_𝑘=1^𝑗_𝑘′=1𝜔_𝑘𝜔_𝑘′ℝ𝑥²𝜓(𝑥)𝜓𝑥+𝑘−𝑘^′≲𝑑𝑥_{2𝑗2𝑘=1𝑗𝑘′=1}𝜔_𝑘𝜔_𝑘′≲2^𝑗2𝑗_𝑘=1𝜔²_𝑘, (34) as well as |∑^2𝑗_𝑘=1∑^2𝑗_𝑘′=1𝜔_𝑘𝜔_𝑘′𝑘∫_ℝ𝑥𝜓(𝑥)𝜓(𝑥+𝑘−𝑘^′)𝑑𝑥|≲2^2𝑗∑^2𝑗_𝑘=1𝜔²_𝑘. Hence, 𝐼≲∑^2𝑗_𝑘=1𝜔²_𝑘, which reaches (32).
Define 𝑞(𝑡)∶=𝐹(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)(𝑡). Then 𝑞,𝑞^′∈𝐿(ℝ) and 𝑞 is locally absolutely continuous. Therefore, 𝑞^′(𝑦)=𝑖𝑦𝑞(𝑦)=𝑖𝑦(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)(−𝑦) and _ℝ𝑦^22𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑²𝑦𝑑𝑦=_ℝ𝑞^′𝑦²𝑑𝑦=2𝜋_ℝ𝑞^′(𝑡)²𝑑𝑡. (35) Clearly, 𝑞^′(𝑡)=∑^2𝑗_𝑘=1𝜔_𝑘[𝜓_𝑗𝑘(𝑡)𝜑^′(𝑡)+𝜑(𝑡)(𝑑/𝑑𝑡)𝜓_𝑗𝑘(𝑡)] and ∫_ℝ|𝑞^′(𝑡)|²𝑑𝑡 ≤ 2[∫_ℝ|𝜑(𝑡)∑^2𝑗_𝑘=1𝜔_𝑘(𝑑/𝑑𝑡)𝜓_𝑗𝑘(𝑡)|²𝑑𝑡+∫_ℝ|𝜑^′(𝑡)×∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘(𝑡)|²𝑑𝑡]≲2^−2𝑗𝛽𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼}[∫_ℝ|∑^2𝑗_𝑘=1𝜔_𝑘𝜓(𝑑/𝑑𝑡)×_𝑗𝑘(𝑡)|²𝑑𝑡+2∫_ℝ|∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘(𝑡)|²𝑑𝑡]  thanks to (C1), (C2), and supp𝜓_𝑗𝑘⊆𝑦,𝑦≥2𝜋32^𝑗. (36) Moreover, ∫_ℝ|𝑞^′(𝑡)|²𝑑𝑡≤𝐶2^−2𝑗𝛽𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼}∑^2𝑗_𝑘=1𝜔²_𝑘 because of (32) and the orthonormality of 𝜓_𝑗𝑘. This with (35), (31), and (30) leads to the desired conclusion of Lemma 6.
Two more classical theorems play important roles in our discussions. We list the first one as Lemma 7, which can be found in [7].

Lemma 7 (Varshamov-Gilbert). Let Ω={𝜔=(𝜔₁,…,𝜔_𝑚),𝜔_𝑘∈{0,1}} with 𝑚≥8. Then there exists a subset {𝜔⁽⁰⁾,…,𝜔^(𝑀)} of Ω such that 𝑀≥2^𝑚/8, 𝜔⁽⁰⁾=(0,…,0), and for 𝑗,𝑙=0,1,…,𝑀,𝑗≠𝑙, ^𝑚_𝑘=1𝜔_𝑘^(𝑗)−𝜔_𝑘^(𝑙)≥𝑚8. (37)

Given two probability measures 𝑃 and 𝑄 on a measurable space (𝕏,ℱ), the Kullback divergence of 𝑃 and 𝑄 is defined by 𝐾(𝑃,𝑄)∶=∫ln𝑑𝑃𝑑𝑄𝑑𝑃,if𝑃≪𝑄,+∞,otherwise. (38) Here, 𝑃≪𝑄 stands for 𝑃 absolutely continuous with respect to 𝑄. In that case, 𝐾(𝑃,𝑄)=∫ln(𝑓_𝑃(𝑥)/𝑓_𝑄(𝑥))𝑓_𝑃(𝑥)𝑑𝑥, where the function 𝑓_𝑃(𝑥) denotes the density function of 𝑃. The second classical theorem is taken from [8].

Lemma 8 (Fano). Let (𝕏,ℱ,𝑃_𝑘) be probability measurable spaces and 𝐴_𝑘∈ℱ,  𝑘=0,1,…,𝑚. If 𝐴_𝑘∩𝐴_𝑣=∅ for 𝑘≠𝑣, then sup_{0≤𝑘≤𝑚}𝑃_𝑘𝐴^𝑐_𝑘1≥min2,𝐶₀√𝑚𝑒^−𝙺𝑚, (39) where 𝙺_𝑚∶=inf_{0≤𝑣≤𝑚}(1/𝑚)∑_𝑘≠𝑣𝐾(𝑃_𝑘,𝑃_𝑣), 𝐶₀=𝑒^−3𝑒−1, and 𝐴^𝑐 denotes the complement of a set 𝐴.

Now, we are in the position to state the main theorem in this section.

Theorem 9. Let 𝜑 satisfy (C1) and (C2), and let 𝑓_𝑛(⋅)∶=𝑓_𝑛(𝑌₁,𝑌₂,…,𝑌_𝑛,⋅) be an estimator of 𝑓∈𝐵^𝑠_𝑟,𝑞(𝐿). Then for 𝑠>0, 𝑝∈[1,∞), 𝑞,𝑟∈[1,∞], and 𝑠≥1/𝑟, there exists 𝐶>0 independent of 𝑓_𝑛 such that with 𝑠^′∶=𝑠−(1/𝑟−1/𝑝)₊, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸𝑓_𝑛−𝑓^𝑝_𝑝≥𝐶(ln𝑛)^{−(𝑠′/𝛼)𝑝}. (40)

Proof. Assume that 𝜓 is the Meyer wavelet function, then 𝜓_𝑗𝑘∈𝐵^𝑠_𝑟,𝑞(ℝ). By Lemma 2, 2^{2−𝑗(𝑠+1/2)𝑗}_𝑘=1𝜔_𝑘𝜓_{𝑗𝑘𝐵𝑠𝑟,𝑞}≲2^{−𝑗(𝑠+1/2)}2^{𝑗(𝑠−1/𝑟+1/2)}×^2𝑗_𝑘=1𝜔_𝑘^𝑟1/𝑟≤1, (41) for 𝜔_𝑘∈{0,1}. Furthermore, with the function ℎ₀ defined in Lemma 5, there exists 𝑐₁>0 such that ℎ_𝜔∶=ℎ₀+𝑐₁2^{−𝑗(𝑠+1/2)}∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘≥0 and ‖ℎ_𝜔‖_𝑠𝑟𝑞≤𝐿 due to that Lemma. Define Λ∶=ℎ_𝜔=ℎ₀+𝑐₁2^{2−𝑗(𝑠+1/2)𝑗}_𝑘=1𝜔_𝑘𝜓_𝑗𝑘,𝜔_𝑘∈{0,1}. (42) Then ∫ℎ_𝜔(𝑥)𝑑𝑥=1 because ∫𝜓(𝑥)𝑑𝑥=0 and ∫ℎ₀(𝑥)𝑑𝑥=1.
By Lemma 7, one finds Λ^′∶={ℎ_𝜔(0),ℎ_𝜔(1),…,ℎ_𝜔(𝑀)}⊆Λ with 𝑀≥2^2𝑗/8 and ℎ_𝜔(0)=ℎ₀ such that for 𝜔≠𝜔^′ and ℎ_𝜔,ℎ_𝜔′∈Λ^′, ∑^2𝑗_𝑘=1|𝜔_𝑘−𝜔^′_𝑘|≥2^𝑗/8. It is easy to see that ℎ_𝜔−ℎ_𝜔′𝑝=𝑐₁2^{−𝑗(𝑠+1/2)2𝑗}_𝑘=1(𝜔_𝑘−𝜔^′_𝑘)𝜓_𝑗𝑘𝑝. (43) This with Lemma 1 leads to ‖ℎ_𝜔−ℎ_𝜔′‖_𝑝≳𝑐₁2^{−𝑗(𝑠+1/2)}×2^{𝑗(1/2−1/𝑝)}(∑^2𝑗_𝑘=1|𝜔_𝑘−𝜔^′_𝑘|)^1/𝑝≳𝑐₁2^−𝑗𝑠, and therefore 𝜃_𝑗∶=min_{ℎ𝜔≠ℎ𝜔′∈Λ′}ℎ_𝜔−ℎ_𝜔′𝑝≳𝑐₁2^−𝑗𝑠. (44) Define 𝐴_𝜔∶={‖𝑓_𝑛−ℎ_𝜔‖_𝑝<𝜃_𝑗/2} for ℎ_𝜔∈Λ^′. Then 𝐴_𝜔∩𝐴_𝜔′=∅, when 𝜔≠𝜔^′. Clearly, ℎ_𝜔∗𝜑 is a density function because both ℎ_𝜔 and 𝜑 are density functions. Let 𝑃^𝑛_{ℎ𝜔∗𝜑} be the probability measure on the Lebesgue space (ℝ^𝑛,ℒ) with the density ∏^𝑛_𝑖=1(ℎ_𝜔∗𝜑)(𝑦_𝑖). Then Lemma 8 tells that sup_{ℎ𝜔∈Λ′}𝑃^𝑛_{ℎ𝜔∗𝜑}𝑓_𝑛−ℎ_𝜔𝑝>𝜃_𝑗21≥min2,𝐶₀𝑀^1/2𝑒^−𝙺𝑀. (45)
According to Lemma 5, ℎ_𝜔(𝑥)≲ℎ₀(𝑥) and 𝑃^𝑛_{ℎ𝜔∗𝜑}≪𝑃^𝑛_ℎ0∗𝜑. Moreover, 𝐾(𝑃^𝑛_{ℎ𝜔∗𝜑},𝑃^𝑛_ℎ0∗𝜑) = ∫_ℝ⋉[ln(Π^𝑛_𝑖=1(ℎ_𝜔∗𝜑)(𝑦_𝑖)/Π^𝑛_𝑖=1(ℎ₀∗𝜑)(𝑦_𝑖))]Π^𝑛_𝑖=1(ℎ_𝜔∗𝜑)(𝑦_𝑖)𝑑𝑦₁𝑑𝑦₂⋯𝑑𝑦_𝑛=𝑛∫_ℝln((ℎ_𝜔∗𝜑)(𝑦)/(ℎ₀∗𝜑)(𝑦))(ℎ_𝜔∗𝜑)(𝑦)𝑑𝑦=𝑛∫_ℝln[1+𝑐₁2^{−𝑗(𝑠+1/2)}(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)(ℎ₀∗𝜑)⁻¹](ℎ_𝜔∗𝜑)(𝑦)𝑑𝑦. Since ℎ_𝜔∗𝜑/ℎ₀ ∗ 𝜑>0, 𝑐₁2^{−𝑗(𝑠+1/2)}(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)(ℎ₀∗𝜑)⁻¹=(ℎ_𝜔∗𝜑−ℎ₀∗𝜑)/ℎ₀∗𝜑>−1. Combining this with ln(1+𝑥)≤𝑥(𝑥>−1), one knows 𝐾𝑃^𝑛_{ℎ𝜔∗𝜑},𝑃^𝑛_ℎ0∗𝜑≤𝑛𝑐₁2^{−𝑗(𝑠+1/2)}×_ℝℎ₀∗𝜑⁻¹𝑦×^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘𝑦×ℎ∗𝜑_𝜔∗𝜑𝑦𝑑𝑦. (46) Because ∫_ℝ(𝜓_𝑗𝑘∗𝜑)(𝑦)𝑑𝑦=𝐹(𝜓_𝑗𝑘∗𝜑)(0)=𝜓_𝑗𝑘(0)𝜑(0)=0, the above inequality reduces to 𝐾(𝑃^𝑛_{ℎ𝜔∗𝜑},𝑃^𝑛_ℎ0∗𝜑)≤𝑐²₁𝑛2^{−2𝑗(𝑠+1/2)}∫[(∑^2𝑗_𝑘=1𝜔_𝑘𝜓_𝑗𝑘∗𝜑)²(𝑦)][(ℎ₀∗𝜑)(𝑦)]⁻¹𝑐𝑑𝑦≲₂₁𝑛2^{−2𝑗(𝑠+1/2+𝛽)}exp{−2𝑐((2𝜋/3)2^𝑗)^𝛼}∑^2𝑗_𝑘=1𝜔²_𝑘≤𝑐²₁𝑛2^{−2𝑗(𝑠+𝛽)}×exp{−2𝑐((2𝜋/3)2^𝑗)^𝛼} thanks to Lemma 6. Hence, 𝙺_𝑀∶=inf_{0≤𝑣≤𝑚}1𝑚_𝑘≠𝑣𝐾𝑃_𝑘,𝑃_𝑣≲𝑐²₁𝑛2^{−2𝑗(𝑠+𝛽)}exp−2𝑐2𝜋32^𝑗𝛼. (47)
Note that 𝑀≥2^2𝑗/8 and take 𝑗 such that 𝑐⁻¹2𝜋3^−𝛼ln𝑛<2^𝑗𝛼≤2^𝛼𝑐⁻¹2𝜋3^−𝛼ln𝑛. (48) Then 𝑀^1/2𝑒^−𝙺𝑀≥exp{((ln2)/16)(3/2𝜋)𝑐^−1/𝛼(ln𝑛)^1/𝛼}exp{−𝐶𝑐²₁𝑛⁻¹[𝑐^−1/𝛼(3/2𝜋)(ln𝑛)^1/𝛼]^{−2(𝑠+𝛽)}}≳1 (choose 𝑐₁>0 small enough). Furthermore, (45) reduces to sup_{ℎ𝜔∈Λ′}𝑃^𝑛_{ℎ𝜔∗𝜑}𝑓_𝑛−ℎ_𝜔𝑝>𝜃_𝑗2≳1. (49) Hence, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸‖𝑓_𝑛−𝑓‖^𝑝_𝑝≥sup_{ℎ𝜔∈Λ′}𝐸‖𝑓_𝑛−ℎ_𝜔‖^𝑝_𝑝≥sup_{ℎ𝜔∈Λ′}𝑃^𝑛_{ℎ𝜔∗𝜑}{‖𝑓_𝑛−ℎ_𝜔‖_𝑝>𝜃_𝑗/2}(𝜃_𝑗/2)^𝑝≳𝜃^𝑝_𝑗. This with (44) and (48) leads to sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸𝑓_𝑛−𝑓^𝑝_𝑝≳(ln𝑛)^{−𝑠𝑝/𝛼}, (50) which is the desired conclusion of Theorem 9, when 𝑟≥𝑝 (𝑠^′=𝑠 in that case).
When 𝑟<𝑝, 𝑠^′=𝑠−(1/𝑟−1/𝑝)₊=𝑠−1/𝑟+1/𝑝, it remains to show sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸𝑓_𝑛−𝑓^𝑝_𝑝≥𝐶(ln𝑛)^{−((𝑠−1/𝑟+1/𝑝)/𝛼)𝑝}. (51) Similar to the proof of (50), one takes small 𝑐₂>0 such that ℎ_𝑘(𝑥)∶=ℎ₀(𝑥)+𝑐₂2^{−𝑗(𝑠−1/𝑟+1/2)}𝜓_𝑗𝑘(𝑥)𝑘=1,…,2^𝑗 (52) satisfies ℎ_𝑘(𝑥)≥0, ℎ_𝑘∈𝐵^𝑠_𝑟,𝑞(𝐿) and ∫ℎ_𝑘(𝑥)𝑑𝑥=1. Clearly, ‖ℎ_𝑘−ℎ₀‖_𝑝=𝑐₂2^{−𝑗(𝑠−1/𝑟+1/2)}2^{𝑗(1/2−1/𝑝)}‖𝜓‖_𝑝 and ‖ℎ_𝑘−ℎ_𝑘′‖_𝑝=𝑐₂2^{−𝑗(𝑠−1/𝑟+1/𝑝)}‖𝜓(⋅)−𝜓(⋅−(𝑘^′−𝑘))‖_𝑝 for 1≤𝑘≠𝑘^′≤2^𝑗. Since 𝜓 is the Meyer wavelet function, inf_𝑘≠0‖𝜓(⋅)−𝜓(⋅−𝑘)‖_𝑝>0 and 𝛿_𝑗∶=min_{0≤𝑘≠𝑘′≤2𝑗}ℎ_𝑘−ℎ_𝑘′𝑝≳𝑐₂2^{−𝑗(𝑠−1/𝑟+1/𝑝)}. (53)
Define 𝐴_𝑘∶={‖𝑓_𝑛−ℎ_𝑘‖_𝑝<𝛿_𝑗/2}(𝑘=0,1,…,2^𝑗). Then 𝐴_𝑘∩𝐴_𝑣=∅(𝑘≠𝑣) and sup_{𝑘=0,1,…,2𝑗}𝑃^𝑛_{ℎ𝑘∗𝜑}𝑓_𝑛−ℎ_𝑘𝑝≥𝛿_𝑗21≥min2,𝐶₀2^𝑗/2𝑒^−𝙺2𝑗, (54) due to Lemma 8. Similar (even simpler) arguments to the estimation of 𝙺_𝑀 show 𝙺₂^𝑗 ≲ 𝑐²₂𝑛2^{−2𝑗(𝑠−1/𝑟+𝛽+1/2)}𝑒^{−2𝑐((2𝜋/3)2𝑗)𝛼}. Taking 𝑗 as in (48), one receives that 𝙺₂^𝑗≤𝐶𝑐²₂𝑛⁻¹𝑐⁻¹2𝜋3^−𝛼ln𝑛^{−(2(𝑠−1/𝑟+𝛽+1/2))/𝛼}, (55) and 2^𝑗/2𝑒^−𝙺2𝑗≥[𝑐⁻¹(2𝜋/3)^−𝛼ln𝑛]^1/2𝛼exp{−𝐶𝑐²₂[𝑐⁻¹(2𝜋/3)^−𝛼ln𝑛]^{−(2(𝑠−1/𝑟+𝛽+1/2))/𝛼}𝑛⁻¹}≥1 by choosing small 𝑐₂>0. Thus (54) reduces to sup_{𝑘=0,1,…,2𝑗}𝑃^𝑛_{ℎ𝑘∗𝜑}𝑓_𝑛−ℎ_𝑘𝑝≥𝛿_𝑗2≳1. (56) Moreover, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸^𝑛_𝑓∗𝜑‖𝑓_𝑛−𝑓‖^𝑝_𝑝≥sup_{𝑘=0,1,…,2𝑗}𝐸^𝑛_{ℎ𝑘∗𝜑}‖𝑓_𝑛−ℎ_𝑘‖^𝑝_𝑝 ≥ sup_{𝑘=0,1,…,2𝑗}𝑃^𝑛_{ℎ𝑘∗𝜑}{‖𝑓_𝑛−ℎ_𝑘‖_𝑝 ≥ 𝛿_𝑗/2}(𝛿_𝑗/2)^𝑝≳𝛿^𝑝_𝑗. This with (53) and (48) leads to (51). This completes the proof of Theorem 9.

Remark 10. By Theorems 9 and 3, the linear wavelet estimator 𝑓^𝐿_𝑛 is optimal for a density in Besov spaces with severely ill-posed noise. Therefore, we do not need to consider nonlinear wavelet estimations in that case. This contrasts sharply with moderately ill-posed noise case under which nonlinear wavelet estimation improves the linear one [2, 4].

Remark 11. When 𝑝=2 and 1≤𝑟≤2, our Theorem 9 is better than Theorem 6 in [2], because (ln𝑛)^{−𝑠′/𝛼}>(ln𝑛)^−𝑠/𝛼. Moreover, Theorems 9 and 3 lead to Theorem 3 in that paper for 𝑝=2 and 1≤𝑟≤2. In addition, our conditions (C1) and (C2) are a little weaker than the assumptions in [2].

4. Concluding Remarks

This paper provides an 𝐿^𝑝(1≤𝑝<∞) risk upper bound for a linear wavelet estimator 𝑓^𝐿_𝑛 (Theorem 3), which turns out to be optimal (Theorem 9). Therefore, nonlinear estimations are not needed under severely ill-posed noises. Although we assume 𝑝<∞ in Theorem 9, the proof of that theorem shows that, for 𝑝=∞, sup_{𝑓∈𝐵𝑠𝑟,𝑞(𝐿)}𝐸𝑓_𝑛−𝑓_∞≥𝐶(ln𝑛)^{−(𝑠−1/𝑟)/𝛼}. (57) In particular, when 𝑟=𝑞=∞, this above estimation reduces to partial result of Theorem 1 in [3].

Note that our model assumes the noise to be severely ill-posed; that is, the density 𝜑 of noise 𝜖 satisfies |𝜑(𝑡)|∼(1+|𝑡|²)^−𝛽/2𝑒^{−𝑐|𝑡|𝛼}(a.e.). Then it is reasonable to choose the Meyer scaling function as 𝜙 because the compact supportness of 𝜙 makes 𝐊_𝑗𝜙 well defined, where 𝐊_𝑗𝜙𝑦1∶=2𝜋_ℝ𝑒^𝑖𝑡𝑦𝜙(𝑡)𝜑−2^𝑗𝑡𝑑𝑡. (58)

Compare with the proof of Theorem 1 in [3], the arguments of Theorem 9 are more complicated in the sense that we use Varshmov-Gilbert Lemma (Lemma 7). It is reasonable because we deal with unmatched estimation sup_{𝑓∈𝐵𝑠𝑟,𝑞(ℝ)}𝐸‖𝑓_𝑛−𝑓‖^𝑝_𝑝 (𝑝 and 𝑟 may not be equal), while they do the matched case sup_{𝑓∈𝐵𝑠∞,∞(ℝ)}𝐸‖𝑓_𝑛−𝑓‖_∞.

Although the Shannon function 𝜙^𝑆(𝑡)=sin𝜋𝑡/𝜋𝑡 is much simpler than the Meyer's, it cannot be used in our discussion because the Shannon’s does not belong to 𝐿(ℝ), while our theorems cover the case for 𝑝=1.

Finally, it should be pointed out that we assume the independence of observations 𝑌₁,𝑌₂,…,𝑌_𝑛 in this paper. However, some dependent data are more important (of course, more complicated) in practice. We will investigate that case in future.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (no. 11271038).