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Volume 12
Issue 18
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Fractional Calculus for Non-Discrete Signed Measures

by
Vassili N. Kolokoltsov
1,2,* and
Elina L. Shishkina
3,4
1
Faculty of Computation Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
2
Moscow Center of Fundamental and Applied Mathematics, 119234 Moscow, Russia
3
Department of Mathematical and Applied Analysis, Voronezh State University, 394018 Voronezh, Russia
4
Department of Applied Mathematics and Computer Modeling, Belgorod State National Research University (BelGU), 308015 Belgorod, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(18), 2804; https://doi.org/10.3390/math12182804
Submission received: 2 August 2024 / Revised: 29 August 2024 / Accepted: 4 September 2024 / Published: 10 September 2024
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
Versions Notes

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. This 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 general fractional calculus; fractional integral with signed measure; fractional power of first-order partial differential operator; quantum mechanic; fractional Poisson brackets; fractional Heisenberg brackets

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MDPI and ACS Style

Kolokoltsov, V.N.; Shishkina, E.L. Fractional Calculus for Non-Discrete Signed Measures. Mathematics 2024, 12, 2804. https://doi.org/10.3390/math12182804

AMA Style

Kolokoltsov VN, Shishkina EL. Fractional Calculus for Non-Discrete Signed Measures. Mathematics. 2024; 12(18):2804. https://doi.org/10.3390/math12182804

Chicago/Turabian Style

Kolokoltsov, Vassili N., and Elina L. Shishkina. 2024. "Fractional Calculus for Non-Discrete Signed Measures" Mathematics 12, no. 18: 2804. https://doi.org/10.3390/math12182804

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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