Talks given at NuHAG events

Deconvolution Based Analysis of Perturbed Integer Sampling in Shift-Invariant Spaces

  Niklas Grip
    Department of Mathematics, LuleĆ„ University of Technology

  given at  strobl07 (17.06.07)
  id:  613
  length:  min
  status:  accepted
  type:  poster
An important cornerstone of both wavelet and sampling theory is _shift-invariant_spaces_, that is, spaces spanned by a Riesz basis of integer-translates of a single function phi, which is referred to as _interpolating_ if phi(n)=delta_{0,n} for integers n.

Under some mild differentiability and decay assumptions on The Fourier transform phi^, we show that \varphi is interpolating and generates a shift-invariant space V if and only if there is a deconvolution phi =g*chi_{[-pi,pi]} for a certain function g with integral one.

Further, we exploit this fact in combination with analysis techniques introduced in a previous paper to derive jitter bounds epsilon = sup_k |epsilon_k| for which any f in V can be reconstructed from perturbed integer samples f(k+epsilon_k).

Finally, we demonstrate the resulting sampling theorem, for example, for some Meyer-type phi and for compactly supported positive g with bounded variation.

The presented results are joint work with Stefan Ericsson.

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