Prolate spheroidal wave functions, quadrature, interpolation, and asymptotic formulae

Whenever physical signals are measured or generated, the results tend to be band-limited (i.e. to 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 band-limited functions are defined on the whole line (or on the 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. They built the requisite analytical apparatus in a sequence of famous papers, and applied the resulting tools in many areas of signal processing, statistics, antenna theory, etc. However, their efforts have not lead to numerical techniques; the principal reason appears to be the lack at the time of effective numerical algorithms for the evaluation of PSWFs and related quantities.; In this dissertation, we start with noticing that in the modern numerical environment, evaluation of PSWFs presents no serious difficulties, and construct a straightforward procedure for the numerical evaluation of PSWFs. Then we use PSWFs to build analogues for band-limited functions of some of the classical numerical techniques: Gaussian quadratures and corresponding interpolation formulae (both exact on certain classes of band-limited functions). We also construct a new class of asymptotic formulae for PSWFs. Unlike the classical apparatus based on Legendre polynomials, our approach utilizes Hermite polynomials, and is valid when bandwidth is large. We illustrate our results with numerical examples.