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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.

Author

Yamada, Akira

Published

2024

Language

EN