# Splines and Fractional Differential Operators

## TL;DR

This paper explores generalized B-splines of complex and hypercomplex orders, deriving associated fractional differential operators and analyzing their action on distribution spaces, expanding the mathematical framework of spline theory.

## Contribution

It introduces fractional differential operators linked to generalized B-splines of complex and hypercomplex orders, extending classical spline theory.

## Key findings

- Derived fractional differential operators for generalized B-splines.
- Identified distribution spaces where these operators act.
- Extended classical spline theory to complex and hypercomplex orders.

## Abstract

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly = 0$ where $L$ is a linear differential operator of integral order. (Cf., for instance, \cite{akhiezer,Golomb,Krein,micchelli,schoenberg}.) In this article, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order, and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.