Estimation of dimension functions of band-limited wavelets

小波变换下带限函数的相位恢复

Phase retrieval of bandlimited functions for the wavelet transform

研究利用小波变换绝对值恢复平方可积信号，证明有限消失矩小波可唯一恢复实值带限信号，并首次提出复值小波采样相位恢复的唯一性结果。

A Class of non-MRA Band-limited Wavelets

We give a characterization of a class of band-limited wavelets of $L^2({\mathbb R})$ and show that none of these wavelets come from a multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$ of ${\mathbb R}$ which is symmetric with respect to the origin. We give necessary and sufficient conditions on a function $ψ\in L^2({\mathbb R})$ with supp $\hatψ\subseteq S_n$ to be an orthonormal wavelet. This result generalizes the characterization of a class of wavelets of E. Hernández and G. Weiss. The dimension functions associated with these wavelets are also computed explicit…

Estimation of wavelet coefficients on some classes of functions

Let $Ψ_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$ W_{2,p}^k=\{f \in L_2(R):\|(I ω)^k\hat f(ω)\|_p\leq 1\}, \, k \in N. $$ We prove that $$ \lim_{m \to \infty}\, \sup\left\{\frac{|(Ψ_m^D)|}{\|(\hat Ψ_m^D)\|_q}: f \in W_{2, p'}^k\right\}=\frac{\frac{(2π)^{1/p-1/2}}{π^k}\left(\frac{1-2^{1-pk}}{pk-1}\right)^{1/p}}{(2π)^{1/q-1/2}}. $$

超越Gevrey正则性的带限小波

Band-limited wavelets beyond Gevrey regularity

本文弱化Gevrey型衰减条件，构造了与扩展Gevrey类相关的次指数衰减正交小波，并用Lambert W函数刻画其精确渐近行为。

Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$

Given a real, expansive dilation matrix we prove that any bandlimited function $ψ\in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame generated by a finite linear combination of dilations of $ψ$ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction relies on a technical condi…

On orthogonal $p$-adic wavelet bases

A variety of different orthogonal wavelet bases has been found in L_2(R) for the last three decades. It appeared that similar constructions also exist for functions defined on some other algebraic structures, such as the Cantor and Vilenkin groups and local fields of positive characteristic. In the present paper we show that the situation is quite different for the field of $p$-adic numbers. Namely, it is proved that any orthogonal wavelet basis consisting of band-limited (periodic) functions is a modification of Haar basis. This is a little bit unexpected because from the wavelet theory point…