Factorizing Entire Functions of Exponential Type

Abstract

The nonlinear problem of factorizing a distribution having compact planar support as a convolution product given possible a priori information about the factors includes the problems of blind deconvolution and signal recovery from magnitude. By the Paley-Wiener-Schwartz theorem [1, p. 390] this problem is equivalent to factorizing entire functions of exponential type. Moreover, typical a priori information about the convolution factors can be equivalently specified in terms of the corresponding entire function. For example, Bochner’s theorem [1, p. 715] implies the correspondence between positive measures and positive definite functions and the Plancherel-Polya theorem [2, p. 353] together with the result in [3, p. 7] implies that the convex closure of the support of a distribution having compact planar support is completely characterized by the growth rate, in various directions, of the corresponding entire function.