Supports of Functions and Integral Transforms

In this paper we apply a method of spectral theory of linear operators [10] to establish relations between the support of a function f on R with properties of its image Tf under a linear operator T : R → R. The classical approach uses analytic continuation of the image Tf into some complex domain (theorems of Paley-Wiener type [4, 5, 6, 7]), and therefore, could not apply to functions whose images Tf are not analytic in any domain (for example, Fourier transforms of functions vanishing on some ball are in general not analytic anywhere). Our approach can be applied to a quite wide class of linear operators, for example, linear integral operators with theorems of Plancherel type, whose kernels are eigenfunctions of some differential operators (Fourier, Hankel, Kontorovich-Lebedev, Yand Htransforms etc. are having these properties). Here we will apply this method only to the Fourier and Hankel transforms. Fourier transforms of compactly supported functions, functions with polynomial domain supports, functions vanishing on some ball and functions vanishing on a half-line are considered. Some of these cases could not be described by the classical method, and even in classical cases (Fourier transforms of compactly supported functions and functions vanishing on a half-line) the results obtained here are also new. Similar results, obtained by different technique, but only for Fourier transforms of functions with bounded supports are considered in [2, 3]. Hankel transform of compactly supported functions is also considered. Applications to the Y-integral transform will be considered in the forthcoming work [9].