Wavelet sampling theorems for irregularly sampled signals

SUMMARY A formula for recovering the original signal from itsirregularly sampled values using wavelets, which extendsthe Walter sampling theorem to the irregular sampling caseand generalizes the PaleyŒWiener 1/4-Theorem by remov-ing the symmetricity constraint for sampling, is pre-sented.a 1999 Scripta Technica, Electron Comm Jpn Pt 3,82(5): 65Œ71, 1999Key words: Wavelet; sampling theorem; scalingfunction; orthogonality; biorthogonality. 1. Introduction In digital signal and image processing, digital com-munications, and so forth, a continuous signal is usuallyrepresented and processed by using its discrete samples.How, then, are we to reconstruct the original signal from itsdiscrete samples? The classical Shannon sampling theoremgives the following formula for band-limited finite energysignals.For a finite energy s-band continuous signalf( t), 2t ˛ R , that is, supp f^(w) I [- s , s] and f ˛ L (R), it canbe recovered by the formulawhere f^ is the Fourier transform of f(t) defined byIf we let s =