Article
Version 1
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Fractianal Calculus for Non-discrete Signed Measures
Version 1
: Received: 6 August 2024 / Approved: 7 August 2024 / Online: 7 August 2024 (17:53:26 CEST)
How to cite: Kolokoltsov, V. N.; Shishkina, E. Fractianal Calculus for Non-discrete Signed Measures. Preprints 2024, 2024080488. https://doi.org/10.20944/preprints202408.0488.v1 Kolokoltsov, V. N.; Shishkina, E. Fractianal Calculus for Non-discrete Signed Measures. Preprints 2024, 2024080488. https://doi.org/10.20944/preprints202408.0488.v1
Abstract
In this paper we suggest a first ever construction of fractional integral
and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann-Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann-Stieltjes integral with a measure having a countable number of extrema. The article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.
Keywords
general fractional calculus; fractional integral with signed measure; fractional power of first order partial differential operator; quantum mechanic; fractional Poisson brackets; fractional Heisenberg brackets
Subject
Physical Sciences, Mathematical Physics
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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