Continuation of Radial Positive Definite Functions and Their Characterization
This 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 f...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/8/623 |
| _version_ | 1797584712238104576 |
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| author | Fethi Bouzeffour |
| author_facet | Fethi Bouzeffour |
| author_sort | Fethi Bouzeffour |
| collection | DOAJ |
| description | This 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. |
| first_indexed | 2024-03-10T23:55:29Z |
| format | Article |
| id | doaj.art-9b0b5ccf1ed14ed1bab5e152a0c6075c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T23:55:29Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-9b0b5ccf1ed14ed1bab5e152a0c6075c2023-11-19T01:11:45ZengMDPI AGFractal and Fractional2504-31102023-08-017862310.3390/fractalfract7080623Continuation of Radial Positive Definite Functions and Their CharacterizationFethi Bouzeffour0Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaThis 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.https://www.mdpi.com/2504-3110/7/8/623positive definite functionscompletely monotone functionsfractional integral and derivativefractional diffusion equation |
| spellingShingle | Fethi Bouzeffour Continuation of Radial Positive Definite Functions and Their Characterization Fractal and Fractional positive definite functions completely monotone functions fractional integral and derivative fractional diffusion equation |
| title | Continuation of Radial Positive Definite Functions and Their Characterization |
| title_full | Continuation of Radial Positive Definite Functions and Their Characterization |
| title_fullStr | Continuation of Radial Positive Definite Functions and Their Characterization |
| title_full_unstemmed | Continuation of Radial Positive Definite Functions and Their Characterization |
| title_short | Continuation of Radial Positive Definite Functions and Their Characterization |
| title_sort | continuation of radial positive definite functions and their characterization |
| topic | positive definite functions completely monotone functions fractional integral and derivative fractional diffusion equation |
| url | https://www.mdpi.com/2504-3110/7/8/623 |
| work_keys_str_mv | AT fethibouzeffour continuationofradialpositivedefinitefunctionsandtheircharacterization |