Positivity of Some Integral Transforms, and Generalization of Bochner's Theorem on Functions of Positive Type

TL;DR

This paper generalizes Bochner's theorem on Fourier transforms of positive type functions by establishing a broader positivity criterion for integral transforms, using solutions of the Schrödinger equation within inverse scattering theory.

Contribution

It introduces a new theorem on the positivity of certain integral transforms and extends Bochner's theorem to more general transforms based on Schrödinger equation solutions.

Findings

Generalized positivity conditions for integral transforms.

Extended Bochner's theorem to broader classes of transforms.

Proved positivity of Fourier cosine transforms of phase-shifts.

Abstract

Using the integral representations of the solutions of Schr\"odinger equation, which are the essential ingredients of the Gel'fand-Levitan and Marchenko integral equations of inverse scattering theory, we obtain a general theorem on the positivity of some integral transforms, and extend the theorem of Bochner on Fourier transforms of functions of positive type to more general transforms. The present study is restricted to the positive half-axis. We then obtain a theorem on the positivity of Fourier cosine transform of the phase-shifts.