On the Supports of Functions

In this paper we establish a relation between the support of a function f on Ω1 ⊂ R with differentiability properties of its image Tf on Ω2 ⊂ R under a linear operator T . The classical approach requires analytic continuation of the image Tf from Ω2 into the complex domain C (theorems of Paley-Wiener type [5, 6, 7]), and therefore, could not apply to functions whose images Tf are not analytic anywhere (for example, the Fourier transform of a function vanishing on some disk is in general not analytic). The transmutation operator approach proposed in Theorem 1 can be applied to investigate supports of functions without passing to the complexification for quite wide class of linear operators. We demonstrate applications of the method to the Fourier transform. The Fourier transform of functions with polynomial domain supports, of functions vanishing on some ball, and of functions vanishing on a half-line is considered. Some of these cases could not be described by the classical way, and even in the classical case (the Fourier transform of functions vanishing on a half-line) the result obtained here is also new. Similar results, obtained by a different technique, but only for the Fourier transform of functions with only bounded supports are considered in [2, 3, 4]. Applications of the method to the Y-transform, the Hankel transform, the Airy transform, and the Mellin transform are demonstrated in [9, 10, 11, 12]. The method can also be applied to investigate such unitary linear integral operators, whose kernels are eigenfunctions of some differential operators.