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

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 ≤ ∞,

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)

  $$;

• (ii)

  $$, where P_jf : = ∑_k∈ℤα_jkϕ_jk;

• (iii)

  $$.

Theorem 3. Let φ satisfy (C3) and ϕ be the Meyer scaling function. If p ∈ [1, ∞), q, r ∈ [1, ∞], $$, then, with $$, (3/8π)((ln n)/4c) ^1/α   <   2^j ≤ (3/4π)((ln n)/4c) ^1/α, s^′ : = s − (1/r − 1/p) ₊, and x₊ = max {x, 0},

In particular, $$ can be replaced by $$, 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 $$. Hence,

When r > p, one obtains that, for some C > 0, $$ and In fact, $$ and Hölder inequality imply that $$ due to |supp  f| ≤ M. By the definition of Besov norm, ∥f∥_spq ≤ C∥f∥_srq. According to (13) and (14), it is sufficient to prove for the conclusion of Theorem 3.

Recall that $$ and $$. Then

By $$ and Lemma 2, To estimate the middle term of (16), one observes that α_jk = ∫_ℝ ϕ_jk(x)f(x)dx, |k²α_jk|   ≤   ∫_ℝ  | 2^jx − k − 2^jx|² | ϕ_jk(x) | f(x)dx≲∫_ℝ  | 2^jx − k|² | ϕ_jk(x) | f(x)dx + ∫_ℝ  | 2^jx|² | ϕ_jk(x) | f(x)dx. Since ϕ is the Meyer scaling function, sup _x∈ℝ | x|² | ϕ(x)| < ∞ and On the other hand, $$ with 1/μ   +   1/μ^′ = 1. Therefore $$ and $$  ≲  $$  ≲  $$. This with Lemma 1 leads to

Now, it remains to consider $$: Using $$ and Lemma 1, one knows

Clearly, $$. Define X_i,k : = (K_jϕ) _jk(Y_i) − E(K_jϕ) _jk(Y_i). Then EX_i,k = 0 and $$. 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)   ×   $$ due to (C3) and $$. Then

Because X_i,k are i.i.d, the Rosenthal′s inequality tells that

This with (21) implies that, for p ≥ 2, $$   >   2}   +   $$. Moveover, (20) reduces to Then it follows from (16)–(19) and (23) that By the choices $$ and (3/8π)((ln n)/4c) ^1/α < 2^j ≤ (3/4π)((ln n)/4c) ^1/α (stated in Theorem 3), one receives that 2^−jps≲(ln n) ^−(s/α)p, 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 $$ over Besov space $$ 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 $$ 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.

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 $$. If ψ is the Meyer wavelet function and |λ_k| ≤ d2^−j/2  (k = 1,2, …, 2^j), then, for some small d > 0,

Proof. It is easy to see that $$ (for r ≥ 1) and $$ by the definition of Besov space. Since

where ⌊s⌋ denotes the largest integer no more than $$ 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 $$ because ψ is the Meyer function. Note that $$   −   2^−jk|²) ⁻¹   ≤   $$. Then for some small d > 0 and |x | ≥ 2,

Hence, (26) holds for |x | ≥ 2. On the other hand, when |x | < 2, h₀(x) ≳ 1 and $$   ≤   $$. Therefore, (26) is true, when d > 0 small enough. This completes the proof of Lemma 5.

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

Proof. As shown in proof of Theorem 1 of [3], one finds easily that (h₀*φ)(y)≳(1 + y²) ⁻¹ and therefore

By Parseval identity, (C1) and $$, $$   =   $$. Moreover, the orthonormality of $$ concludes that

To estimate $$, one proves an inequality:

Note that $$,  $$, and $$   =   $$. Then $$   ×   $$   +   k) ²ψ(x)ψ[x − (k^′ − k)]dx and Since 〈ψ(x), ψ(x − k)〉 = δ_k,0, $$; On the other hand, the boundedness of ∫_ℝ x²ψ(x)ψ(x + l)dx and ∫_ℝ xψ(x)ψ(x + l)dx implies that as well as $$. Hence, $$, which reaches (32).

Define $$. Then q, q^′ ∈ L(ℝ) and q is locally absolutely continuous. Therefore, $$ and

Clearly, $$ and ∫_ℝ  | q^′(t)|²dt   ≤   $$  thanks to (C1), (C2), and Moreover, $$ 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,

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

where $$, $$, and A^c denotes the complement of a set A.

Theorem 9. Let φ satisfy (C1) and (C2), and let f_n(·): = f_n(Y₁, Y₂, …, Y_n, ·) be an estimator of $$. 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) ₊,

Proof. Assume that ψ is the Meyer wavelet function, then $$. By Lemma 2,

for ω_k ∈ {0,1}. Furthermore, with the function h₀ defined in Lemma 5, there exists c₁ > 0 such that $$ and $$ due to that Lemma. Define Then ∫ h_ω(x)dx = 1 because ∫ ψ(x)dx = 0 and ∫ h₀(x)dx = 1.

By Lemma 7, one finds $$ with $$ and $$ such that for ω ≠ ω^′ and $$, $$. It is easy to see that

This with Lemma 1 leads to $$, and therefore Define $$ for h_ω ∈ Λ^′. Then $$, when ω ≠ ω^′. Clearly, h_ω*φ is a density function because both h_ω and φ are density functions. Let $$ be the probability measure on the Lebesgue space (ℝⁿ, ℒ) with the density $$. Then Lemma 8 tells that

According to Lemma 5, h_ω(x)≲h₀(x) and $$. Moreover, $$   =   $$. Since h_ω*φ/h₀  *  φ > 0, $$. Combining this with ln (1 + x) ≤ x(x > −1), one knows

Because $$, the above inequality reduces to $$ thanks to Lemma 6. Hence,

Note that $$ and take j such that

Then $$ (choose c₁ > 0 small enough). Furthermore, (45) reduces to Hence, $$. This with (44) and (48) leads to 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

Similar to the proof of (50), one takes small c₂ > 0 such that satisfies h_k(x) ≥ 0, $$ and ∫ h_k(x)dx = 1. Clearly, $$ and $$ for 1 ≤ k ≠ k^′ ≤ 2^j. Since ψ is the Meyer wavelet function, inf _k≠0∥ψ(·)−ψ(·−k)∥_p > 0 and

Define $$. Then A_k∩A_v = ∅  (k ≠ v) and

due to Lemma 8. Similar (even simpler) arguments to the estimation of $$ show $$  ≲  $$. Taking j as in (48), one receives that and $$ by choosing small c₂ > 0. Thus (54) reduces to Moreover, $$   ≥   $$   ≥   $$. 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 $$ 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 $$. 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].