Fractional Calculus for Non-Discrete Signed Measures

Fractional Calculus for Non-Discrete Signed Measures

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 m...

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Bibliographic Details
Main Authors: Vassili N. Kolokoltsov, Elina L. Shishkina
Format: Article
Language:English
Published: MDPI AG 2024-09-01
Series:Mathematics
Subjects:
general fractional calculus
fractional integral with signed measure
fractional power of first-order partial differential operator
quantum mechanic
fractional Poisson brackets
fractional Heisenberg brackets
Online Access:https://www.mdpi.com/2227-7390/12/18/2804
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Summary: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.
ISSN:2227-7390

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