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Continuation of Radial Positive Definite Functions and Their Characterization
Version 1
: Received: 25 July 2023 / Approved: 10 August 2023 / Online: 11 August 2023 (12:54:09 CEST)
A peer-reviewed article of this Preprint also exists.
Bouzeffour, F. Continuation of Radial Positive Definite Functions and Their Characterization. Fractal Fract. 2023, 7, 623. Bouzeffour, F. Continuation of Radial Positive Definite Functions and Their Characterization. Fractal Fract. 2023, 7, 623.
Abstract
TThis paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann-Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.
Keywords
Positive definite functions; completely monotone functions; Fractional integral and derivative; Fractional diffusion equation
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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