Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Haizhen Li,Xiaojun Fan ,Yan Tang

doi:10.1186/s13660-021-02712-w

Abstract

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of L_{p,{1 / omega }}-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Highlights

1 Introduction The classical Shannon sampling theorem shows that a bandlimited signal which lives in the shift-invariant space generated by the sinc function can be recovered from its samples {f}n∈Z when the gap δ is small enough [1]
Since the sinc function has infinite support and slow decay, the space of bandlimited functions is often unsuitable for numerical implementations
We mainly study the asymptotic pointwise error estimates for signals in Vp,1/ω(φ) under the assumptions (i) φ ∈ W1,∞,ω(Rd). (ii) limδ→0 ωδ(φ) W1,∞,ω = 0, where ωδ(φ) is the continuous modulus defined by ωδ(φ)(x) := sup φ(x + y) – φ(x)

Summary

Introduction

The classical Shannon sampling theorem shows that a bandlimited signal which lives in the shift-invariant space generated by the sinc function can be recovered from its samples {f (nδ)}n∈Z when the gap δ is small enough [1]. The first one is f , ψγ = f (x)ψγ (x) dx, Rd where the average sampling functionals {ψγ : γ ∈ } satisfy the following: (a) Rd ψγ (x) dx = 1 for all γ ∈ . Lemma 2.1 ([7]) Let ω be a submultiplicative weighting function.

Results

Conclusion