On mutually inverse transforms of functions on a half-line

TL;DR

This paper investigates two transforms on functions defined on a half-line, showing their composition yields a concave majorant and acts as the identity on nonnegative concave functions, with applications in mathematical physics.

Contribution

It introduces and analyzes two mutually inverse transforms, establishing their properties and applications, including the identity on nonnegative concave functions.

Findings

Composition of the transforms yields a concave majorant.

The composition acts as the identity on nonnegative concave functions.

Applications to mathematical physics are discussed.

Abstract

Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative concave functions. Applications of this result to some problems of mathematical physics are indicated. Several open questions are formulated.