Finite Fourier self-transforms

Abstract

Abstract.This paper considers the integral equation XY(t') = (2tt)~n f exp (-ia-t') [ 7(t) exp (io>■ t) dt d<oJq J t as well as a more general one wherein the Fourier kernels are weighted. When Q and T are ^-dimensional spherical domains, the eigenfunctions of the integral equation are generalized prolate spheroidal functions for which a new nomenclature is proposed. Many properties of the eigenfunctions are developed and summarized. Because of the importance of these functions in Fourier transform theory, old as well as new properties are included. Introduction.In a series of papers Slepian, Landau and Pollak [1, 2, 3, 4] developed many of the properties of a new set of functions which they called generalized prolate spheroidal wave (GPSW) functions. These functions play an important role in the theory of Fourier transforms. Since Fourier transforms are widely used in many fields, the GPSW functions have found many applications, particularly in linear circuit theory and diffraction. [11][12][13][14][15][16][17][18]23] give a good sampling of such applications; others may be found

Strings in Background Fields

“…Then one can no longer speak of D0 and D1 branes but rather has wave packets, such as Gaussian wave-packets in a non-compact space. As in non-compact space, wave-functions exist which are self-dual under Fourier transforms [30]. In that case one can no longer speak of a fixed brane dimension.…”