On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

Junjian Zhao,Wei-Shih Du,Marko Kostić

doi:10.3390/sym13020331

Abstract

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

Highlights

Introduction and PreliminariesTheorem in Shift-Invariant SubspacesWe first introduce and review some spaces that will be used in this paper
In this article, motivated by the above studies, we explore the promotion of the results known in the literature in shift-invariant subspaces of mixed-norm Lebesgue spaces L~p (Rd )
∑l |η (ξ + 2πl )|2 > 0 for every ξ ∈ Rd. It is well-known that sampling theory and stability theory are fascinating theories that has a wide range of applications in different branches of mathematics

Summary

Introduction

Introduction and PreliminariesTheorem in Shift-Invariant SubspacesWe first introduce and review some spaces that will be used in this paper. The Wiener amalgam spaces (WL) p (Rd ) is the set of all measurable functions g on Rd such that their norms k g( x )k(WL) p (Rd ) defined by k g( x )k(WL) p (Rd ) The following mixed-norm spaces will be discussed in this paper. The mixed-norm discrete space l~p (Zd ) is defined by l~p Z

Results

Conclusion