Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II

Abstract

The theory developed in the preceding paper is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite‐energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.

Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order

“…Taking simply q λ n = sign f n λ does not work because there exist infinitely many independent bandlimited functions ϕ that are everywhere positive (such as the lowest order prolate spheroidal wave functions [16], [14] for arbitrary time intervals and symmetric frequency intervals contained in [−π, π]); picking the signs of samples as candidate q λ n would make it impossible to distinguish between any two functions in this class.…”

Time-Frequency Localization and Sampling of Multiband Signals

“…We begin reviewing the work of Landau, Slepian and Pollak [35,28,29,32] -the so-called Bell Labs theory -concerning "time-and bandlimiting" (see also [22,24,34,26]) in Section 2. The Paley-Wiener space PW consists of those functions in L 2 (R) whose Fourier transforms vanish outside the set [−1/2, 1/2].…”

Wave propagation using bases for bandlimited functions

“…In order to manage this apparent contradiction, it is natural to consider the basis of eigenfunctions of the space and band limiting operator. This has been the topic of a series of papers by Slepian et al [1][2][3][4][5], which introduced the prolate spheroidal wave functions (PSWFs) as an eigensystem bandlimited in [−c, c] and maximally concentrated within the space interval [−1, 1].…”

“…Let us briefly review the results in [1,2,20] relevant to the purposes of this paper. The PSWFs are constructed for a fixed bandlimit c > 0.…”

“…Instead, we consider bandlimited functions on an interval. A basis for bandlimited functions, the prolate spheroidal wave functions (PSWFs), was introduced in the 1960s by Slepian et al in a series of papers [1][2][3][4][5]. Recently the generalized Gaussian quadratures became available in [6,7], making it possible to construct efficient numerical algorithms for such functions.…”