Fractional Operators, Dirichlet Averages, and Splines

TL;DR

This paper explores the deep mathematical connections between fractional operators, Dirichlet averages, and splines of complex order, revealing their interrelated properties and extending their definitions in infinite-dimensional spaces.

Contribution

It establishes fundamental links between three distinct mathematical areas and extends the definitions of Dirichlet averages and splines of complex order to infinite dimensions.

Findings

Unified framework for fractional derivatives and integrals on Lizorkin spaces

Extension of Dirichlet averages to infinite-dimensional settings

Connections between splines of complex order and fractional operators

Abstract

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented.