A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Jianping Zhang,Alberto Fiorenza

doi:10.1155/2022/1372184

Abstract

In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.

Highlights

Let H be a separable Hilbert space
Starting from a pair of general refinable functions without smoothness restrictions, they obtained a construction of weak dual wavelet frames for reducing subspace FL2ðΩÞ of L2ðRdÞ
Inspired by all these works, in present paper, we investigate a class of weak dual wavelet frames for reducing subspaces of Sobolev spaces

Summary

Introduction

Let H be a separable Hilbert space. An at most countable sequence feigi∈I in H is called a frame for H if there exist two constants 0 < C1 ≤ C2 < ∞ such that. We wavelet say ðX frames sfðoψr0ð, FΨHÞ,sXðΩ−sÞð,ψ~F0H, Ψ~−sÞðÞΩiÞsÞ a if pair of weak dual (1) fTkψl : k ∈ Zd, 0 ≤ l ≤ Lg and fTkψ~l : k ∈ Zd, 0 ≤ l ≤ Lg are Bessel sequences in HsðRdÞ and H−sðRdÞ, respectively (2) There exist dense subsets V of FHsðΩÞ and V~ of F H−sðΩÞ such that h f , gi =. Starting from a pair of general refinable functions without smoothness restrictions, they obtained a construction of weak dual wavelet frames for reducing subspace FL2ðΩÞ of L2ðRdÞ. Inspired by all these works, in present paper, we investigate a class of weak dual wavelet frames for reducing subspaces of Sobolev spaces.

The Characterization of Weak Dual Wavelet Frames

Conclusion