Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case

In many branches of technology, such as sampled-data theory, timeseries analysis, etc., doubly infinite sequences of complex numbers, hn} = ..., h~i, ho, hi,... play an important role. Associated with such a sequence is its amplitude spectrum H(f) = t hne*TMf. (1) In this paper we attempt to elucidate certain features of the complex relationship between {/i,,| and its amplitude spectrum H(f). Of prime importance in the analysis we present are some special sequences, here called discrete prolate spheroidal sequences (DPSS's), and some related special functions called discrete prolate spheroidal wave functions (DPSWF's). Much of the paper is devoted to a study of their mathematical properties. They are fundamental tools for understanding the extent to which sequences and their spectra can be simultaneously concentrated: they have many potential applications in communications technology. 1371 We motivate our work by discussing a simple problem. But first some notation is needed. We adopt the abbreviation 12 E(nltn2)s L hn2 (2) n=n 1