Inverses of integral transforms of RKHSs

TL;DR

This paper investigates when the inverse of an integral transform in Hilbert spaces can be expressed with a complex conjugate kernel, providing a new proof of Plancherel's theorem through reproducing kernel theory.

Contribution

It establishes a necessary and sufficient condition for representing the inverse of integral transforms with conjugate kernels, extending Saitoh's work.

Findings

Derived a condition for inverse transforms to have conjugate kernels

Provided an alternative proof of Plancherel's theorem

Extended the theory of integral transforms in RKHSs

Abstract

The Fourier transform and its inverse are well-known to have complex conjugate integral kernels. S.~Saitoh demonstrated that this relationship extends to the theory of integral transforms of Hilbert spaces of functions under certain conditions. In this paper, we derive a necessary and sufficient condition for the inverse of an integral transform of a Hilbert space of functions to be represented by a complex conjugate integral kernel. As an application, we present an alternative proof of Plancherel's theorem using the theory of reproducing kernels.