The zeros of entire functions: Theory and engineering applications

Abstract

Functions that are analytic over a complex plane are called entire functions. They may be viewed as generalizations of polynomials because they admit power series expansions that converge everywhere. Entire functions are useful models for physical phenomena having finite "spectra," such as important classes of time-varying signals and spatially varying fields. Entire function models may be studied through the linear expansion techniques (e.g., Fourier series and integrals) familiar to most engineers, but product expansions akin to polynomial factorizations provide a less familiar alternative which is often more powerful when phenomena have a nonlinear character. This paper provides a tutorial introduction to relevant aspects of the theory of entire functions of a single complex variable, and also a discussion of exemplary applications. The paper deals specifically with a class of entire functions of exponential type, dubbed B-functions, which contain all functions that are band limited according to the variious extant definitions of bandlimitation. Product expansions are emphasized because they lead one to regard the real and complex zeros of a B-function as its information bearing attributes. This view leads naturally to the study of sampling, clipping, and similar applications, and to more general issues concerned with the sufficiency of zero-based representations and the recovery of a B-function's waveform from a zero-based representation.