The Master Integral Transform with Entire Kernels

Mohammad Abu-Ghuwaleh ()
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Mohammad Abu-Ghuwaleh: Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan

Mathematics, 2025, vol. 13, issue 21, 1-27

Abstract: We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks.

Keywords: integral transforms; entire kernels; Beurling–Malliavin density; kernel–signal duality; Mellin inversion; completeness frames (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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