2024/12/26 by Akira Yamada, Yamada, Akira
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Control Systems and Analysis #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2412.19047
openalex publication_date 2024/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.