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Whenever physical signals are measured or generated, the results tend to be band-limited (i.e. have compactly supported Fourier transforms). Indeed, measurements of electromagnetic and acoustic data are band-limited due to the oscillatory character of the processes that have generated the quantities being measured; when the signals being measured come from heat propagation or diffusion processes, they are (practically speaking) band-limited, since the underlying physical processes operate as low-pass filters. The importance of band-limited functions has been recognized for hundreds of years; classical Fourier analysis can be viewed as an apparatus for dealing with such functions. When the latter are defined on the whole line (or on a circle), classical tools are very satisfactory. However, in many cases, we are confronted with band-limited functions defined on intervals (or, more generally, on compact regions in $\R^n$). In this environment, standard tools based on polynomials are often effective, but not optimal. In fact, the optimal approach was discovered more than 30 years ago by Slepian et al, who observed that for the analysis of band-limited functions on intervals, Prolate Spheroidal Wave Functions (PSWFs) are a natural tool. Although they built the requisite analytical apparatus in a sequence of famous papers, few numerical techniques ensued. Apparently, the principal reason for the lack of popularity of PSWFs was the absence of necessary numerical evaluation schemes. In this talk, we will present recent developments in the theory of band-limited functions. we will start with noticing that in the modern numerical environment, evaluation of PSWFs presents no serious difficulties, and present a straightforward procedure for the numerical evaluation of PSWFs and related quantities. Based on PSWFs, we have constructed integration and interpolation schemes (both exact on certain classes of band-limited functions), which are analogous to the classical Gaussian quadratures and corresponding interpolation formulae for polynomials. We will illustrate our results with several examples.

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