Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces |
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Authors: | Stefan Ericsson |
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Affiliation: | Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden |
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Abstract: | 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 for some g with . We explain why analysis of this particular generating function can be more likely to provide large jitter bounds ε such that any f∈V can be reconstructed from perturbed integer samples f(k+εk) whenever supk∈Z|εk|?ε. We use this natural deconvolution of 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 φ 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. |
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Keywords: | Shift-invariant space Reproducing kernel Interpolating function Shift-invariant Deconvolution Irregular sampling Scaling function Shannon wavelet Franklin B-spline Meyer wavelet |
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