Fractional Calculus for Non-Discrete Signed Measures

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