A Jackson-type inequality associated with wavelet bases decomposition

*Correspondence: gtotony98@gmail.com 1School of Information Engineering, Sanming University, No. 25, Jing Dong Road, 365004 Sanming City, Fujian, China Abstract Although wavelet decompositions of functions in Besov spaces have been extensively investigated, those involved with mild decay bases are relatively unexplored. In this paper, we study wavelet bases of Besov spaces and the relation between norms and wavelet coefficients. We establish the lp-stability as a measure of how effectively the Besov norm of a function is evaluated by its wavelet coefficients and the Lp-completeness of wavelet bases. We also discuss wavelets with decay conditions and establish the Jackson inequality.

The unconditional summability is a key problem in characterizing Lebesgue spaces and thus Besov spaces. The authors of [2,3,18] suggest that the smoothness and regularity are necessary for the Besov unconditional summability. This evolves more technicality in the case of Lebesgue spaces. The present author contends that the decay on wavelets suffices for a positive answer. Our approach involves two parts: characterizing Lebesgue spaces by wavelet coefficients and the design for middle class (3.1), which depends only on wavelet coefficients. We will associate the L p -unconditional summability to a special design for equivalent Besov norms. We will see that the l p -stability for functions in a Lebesgue space can provide an alternative framework for Besov-unconditional summability, as well as Jackson inequality. We remark that the use of wavelet bases is motivated by [21,Rem. 2.2], [28, pp. 130, 136], and [30]. However, canonical dual bases without wavelet structures may not contribute to the unconditional summability. This paper is divided into four sections. After this introduction, Sect. 2 collects some preliminary facts, especially about the L p -boundedness of affine operators. Section 3 presents a number of applications of bounded affine operators, including completeness, l p -stability of biorthogonal Riesz wavelet bases, middle class K τ ,τ , the Jackson inequality, and characterization of Besov spaces. Discussion and conclusions are given in Sect. 4.

Preliminaries
We use the following notation: Let X, Y be two quasi-normed spaces. By writing X → Y we mean that X is continuously embedded in Y ; in other words, we can think of X ⊂ Y and · Y · X . A sequence Recall that the Lorentz space l p,q (N), 0 < p, q < ∞ [24, p. 955], consists of sequences a = {a i } i∈N satisfying a l p,q (N) < ∞, where where a * 1 := a l ∞ (N) and a * i := max For p = q, we have l p,q (N) = l p (N) with equivalent norms. The spaces are ordered lexicographically. More precisely, let 0 < p 1 , p 2 < ∞ and 0 < q 1 , q 2 ≤ ∞. Then l p 1 ,q 1 l p 2 ,q 2 if p 1 < p 2 , or if p 1 = p 2 and q 1 < q 2 .
A sequence {f i } i∈Z is said to be l p,q -Hilbertian [19, p. 59] in a Banach space X if i∈Z a i f i converges unconditionally in X and i∈Z a i f i X a l p,q (Z) for any a = {a i } i∈Z ∈ l p,q (Z), 0 < p, q ≤ ∞.
(2) n x n converges for every choice of signs n = ±1. (3) λ n x n converges for every bounded sequence of scalars {λ n }.
for all f ∈ L 2 (R).
Here A and B are called frame bounds. If A = B, then we call this a tight frame. A Riesz basis is a frame that consists of linear independent basic vectors. A sequence {ψ i : i ∈ Z} is a Riesz basis for L 2 (R) if for any {c i } ∈ l ∞ , for some positive constants A, B. The frame operator S of {ψ i : i ∈ Z} is defined by and each f ∈ L 2 (R) has the decomposition The preframe operator T : l 2 (N) → L 2 (R) is a bounded linear operator defined by Note that S = TT * . Both S and S are of type (2, 2), bounded, invertible, self-adjoint, and positive on L 2 (R). The sequence {S ψ i : i ∈ Z} is also a frame for L 2 (R), and its frame operator is S . The canonical dual frame {S ψ i : i ∈ Z} of a tight frame is simply { 1 A ψ i : i ∈ Z}. More details can be found in [8,21].
The affine wavelet frame system {ψ j,k } = {ψ j,k : j, k ∈ Z} for L 2 (R) generated by ψ is defined by In general, [8, p. 276]). In this case, we say that the canonical dual frame {S ψ j,k } of {ψ j,k } does not have wavelet structure.
Two sequences {ψ j,k } and { ψ j,k } form a pair of dual wavelet frames/biframes ([8, p. 277], [9,14]) if both are frames for L 2 (R) and They form a pair of biorthogonal (Riesz) wavelet bases if they are also Riesz bases for L 2 (R). Obviously, orthogonal wavelet bases for L 2 (R) are also biorthogonal Riesz wavelet bases.
A wavelet has zero mean [10, p. 433] and is in L 2 . In addition, compactly supported or exponential decay orthogonal wavelets cannot belong to C ∞ (R) ∩ L 2 (R) ( [17], [23,Sect. 4.6,6]). More precisely, if such a wavelet exists, then it must be the zero function on R. We where (x) := sup 0≤x≤|y| |ψ|(y), and τ > 0. Let L ψ be a constant such that ψ ∈ M τ for all The infinite integrals either both converge or both diverge. If ψ is compactly supported, then ψ belongs to M τ for all τ ∈ (0, ∞).
Denote F ψ,M τ := {ψ j,k : j, k ∈ Z, τ > 0}, which is a frame for L 2 (R) with ψ ∈ M τ . Let F be a sequence in L p (R) and consider the set of all possible m-term expansions with elements from F , We denote the error of the best m-term approximation to f ∈ L p (R) by Proof The boundedness of λ m and λ m follows from the Riesz-Thorin interpolation theorem [16, p. 32]. Let (·) := sup 0≤·≤|y| |ρ|(y). For {b k } k∈Z ∈ l 1 (Z) and 2 j x + k = u + k , we have The finiteness of k ∈Z |ρ|(u + k ), u < 0, can also be obtained by setting Similarly, let us address {b k } k∈Z ∈ l ∞ (Z), and let v = 2 j x. We obtain So λ m is also bounded on l 1 (Z) and l ∞ (Z). By the Riesz-Thorin interpolation theorem, λ m is bounded on l τ (Z) for any 1 < τ < ∞. Similarly, λ m is bounded on l τ (Z) for any 1 ≤ τ ≤ ∞ by using the method described above.
For the statement on , τ ≥ 1, we need to show that Similar arguments give j ,k ∈Z j,k∈Z j>j By the Minkowski inequality, (2.10) holds.
For 0 < τ < 1, we have the following inequality: For the second summation in (2.11), by a similar argument we have j ,k ∈Z j,k∈Z j≤j

Biorthogonal Riesz wavelet bases in Besov spaces
Theorem 3.1 is established in our earlier work [21,Thm. 3.3]. It serves as a base for understanding the L p -unconditional summability. There several consequences of Theorem 3.1. First, the frame and synthesis operators are L p -bounded. Second, biorthogonal Riesz wavelet bases are unconditionally summable in Lebesgue spaces.

Theorem 3.1 Let F ψ,M 1 and F ψ,M 1 be a pair of biorthogonal Riesz wavelet bases for L 2 (R).
(1) The operator S associated with F ψ,M 1 is L p -bounded and bijective on L p (R) for all 1 < p < ∞.
Despite the feasibility of biframes, we paid the price of the linear independence, and we can deduce from Theorem 3.1 that the generator for the canonical dual bases requires mild decay to confirm the bijectivity of the frame operator on L p (R), 1 < p < ∞. Secondly, the conditions for Theorem 3.1 and Theorem 3.2 have been improved significantly in comparison with the results in [2,3,6], [ Here we give two examples. Local commutant biorthogonal bases can be found in [5]. The last example refers to those discovered by Lemvig and Bownik [4,p. 219, Example] and [25] and is a family of band-limited wavelet frames. More information on (bi)orthonormal bases can be found in [21,Sect. 4].
Next, we characterize Lebesgue spaces by wavelet coefficients. For f ∈ L p (R), Theorem 3.2 guarantees { f , ψ (p ) j,k } ∈ l p (Z × Z), which leads characterization of the middle class K τ ,τ (L p (R), F · ) depending only on wavelet coefficients. The middle class is designed to contribute to understanding the relationship between the best m-term approximation and Besov norms. (1) Both primal wavelet bases {ψ j,k } are unconditional and l p,1 -Hilbertian in L p (R) for all 1 < p < ∞. Moreover, for any f = j,k∈Z c j,k ψ f p c j,k l p c j,k l τ for all 1 < p < ∞, τ ∈ (0, p).
Moreover, if ψ and ψ are compactly supported, then with equivalent norms for all τ ∈ (0, p).
Proof (1) We consider Utilizing Theorem 3.1 and the L p -boundedness of R, we obtain the unconditional summability for {ψ j,k } in L p (R). Note that l p = l p,p with equivalent norms and l τ ⊂ l p,1 ⊂ l p,p , τ ∈ (0, p). Let f = j,k∈Z c j,k ψ (p) j,k ∈ L p (R), and let t = 2 j x for a given j. The Hölder inequality for the summation over k yields Thus, by the Minkowski inequality, f p c j,k l p c j,k l p,1 c j,k l τ . Indeed, the finiteness of k∈Z |ψ|(t + k) can be found in Lemma 2.2. Similar arguments can be made for { ψ (p ) j,k }.
(2) Again, applying Theorem 3.1, we see that both F ψ,M 1 and F ψ,M 1 are bases for L p (R), 1 < p < ∞. By (1) and (2.6) we have By the admissible hypotheses and the range of τ we have Finally, the compactly supported cases can be done similarly and are skipped.
(2) Let the reference wavelets (ρ, ρ) satisfy the hypotheses with the following properties. For all g ∈ B α ⊂ L p (R), α = 1 τ -1 p , 1 p + 1 p = 1, 1 < p < ∞, the expansion of g in the reference wavelet system is given by which converges unconditionally with coefficients D := {d j ,k : j , k ∈ Z} satisfying |g| B α ∼ D l τ (Z×Z) for all τ ∈ (0, p). Note that (3.7) With the admissible hypotheses and the range of τ , applying Lemma 2.2 and (3.7), we deduce that Combining (1), (3.8), and Theorem 3.2(2), we see that the Jackson inequality holds: (3) With the admissible hypotheses and the range of τ , applying Theorem 3.2(1) and (3.8), we see that the synthesis operator U and the analysis operator U given by are bounded, and thus UU is bounded. Since B α are dense in both L p (R) and L 2 (R), any function h has a wavelet frame expansion The boundedness of UU guarantees that the sum converges unconditionally in L p (R).