Fractional Calculus for Non-Discrete Signed Measures

Vassili N. Kolokoltsov () and Elina L. Shishkina
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Vassili N. Kolokoltsov: Faculty of Computation Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
Elina L. Shishkina: Department of Mathematical and Applied Analysis, Voronezh State University, 394018 Voronezh, Russia

Mathematics, 2024, vol. 12, issue 18, 1-12

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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