Remark on the theory of reproducing kernels

TL;DR

This paper critically examines the foundations of reproducing kernel Hilbert space theory, challenging previous claims about characterizations of the range of linear transforms in existing literature.

Contribution

It provides a correction to prior claims regarding the characterization of the range of linear transforms in reproducing kernel Hilbert spaces.

Findings

Previous claims about the range characterization are incorrect.

The paper clarifies foundational aspects of reproducing kernel theory.

It refutes specific assertions made in Saitoh's works.

Abstract

Foundations of the theory of Hilbert spaces with reproducing kernels are discussed. It is demonstrated that the claims in the papers of S.Saitoh and in his book "Theory of reproducing kernels and applications, Pitman research notes, 189, Longman, New York, 1988, that a characterization of the range of arbitrary linear transform is obtained in his works, is not correct.