Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:373

Article

Ericsson, Stefan¹  Grip, Niklas¹ 

[1] Lulea Univ Technol, Dept Math, SE-97187 Lulea, Sweden

关键词: Shift-invariant space;    Reproducing kernel;    Interpolating function;    Shift-invariant;    Deconvolution;    Irregular sampling;    Scaling function;    Shannon wavelet;    Franklin;    B-spline;    Meyer wavelet;   

DOI  :  10.1016/j.jmaa.2010.07.021

来源: Elsevier

【 摘 要 】

An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function with Fourier transform (phi) over cap= integral(xi+pi)(xi-pi) g(nu)dv for some g with integral(R) g(xi) = 1. We explain why analysis of this particular generating function can be more likely to provide large jitter bounds epsilon such that any f E V can be reconstructed from perturbed integer samples f (k + epsilon(k)) whenever sup(k epsilon Z)vertical bar epsilon(k)vertical bar <= epsilon. We use this natural deconvolution of (phi) over cap(xi) to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of yo for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free