Time-localized Sampling Approximations

Abstract

In this last chapter we explore briefly some connections among sampling and time and band limiting. The chapter begins by pointing out a general connection between the samples of eigenfunctions of time and band limiting and the eigenvectors of a certain matrix whose entries are, in essence, the samples of time-localized images of functions that interpolate the samples in the given Paley–Wiener space. Next, a discrete method is considered for generating eigenfunctions of time–frequency localizations to unions of sets from their separate localizations. We then reconsider the connection between eigenfunctions and their samples in the concrete context of localization to intervals of the real line, outlining work ofWalter and Shen [347] and of Khare and George [177].Walter and Shen provided L ²-estimates for approximate prolate spheroidal wave functions (PSWFs) constructed from interpolation of their sample values within the time-localization interval. We provide a partial sharpening of their estimates by using a slightly enlarged set of samples.