Irregular sampling of wavelet and short-time Fourier transforms
It is well-known that when an admissible analyzing wavelet is used to define the integral wavelet transform or when an admissible window function is used to formulate the short-time Fourier (or Gabor) transform, then every square-integrable function on the real line can be perfectly recovered from the values of the transform (wavelet or STFT) in the entire time-scale (or time-frequency) plane. This paper is concerned with perfect reconstruction from only discrete samples of the transform. When these discrete values are regular (or gridded), the theory and techniques of frames can be used; but when irregular samples on the time- scale (or time-frequency) plane are considered, the problem becomes exceedingly difficult. This paper is an attempt to establish some preliminary results to this problem. Due to the limitation of the analytic method, only analyzing wavelets and window functions \(g\) that are entire functions of exponential type are used. Under this and other constraints, an iterative scheme is also formulated to reproduce the \(L^ 2\) functions \(f\) that are analyzed. If \(f\) also happens to be bandlimited with bandwidth no larger than the bandwidth \(\Omega\) of \(g\), then it is also shown that the \(L^ 2\)-norm of \(f\) is equivalent to the \(\ell^ 2\)-norm of \(\biggl\{ \bigl[ (x_{k+1}-x_{k-1})/2 \bigr] ^{1/2}f(x_ k) \biggr\}\) provided that the mash width \(\max | x_{k+1}-x_ k |\) is strictly less than \(\pi/ \Omega\).
scientific article; zbMATH DE number 205302
scientific article