Fourier representations of fractional B Splines via generalized Stirling type polynomials

TL;DR

This paper introduces a novel Fourier expansion for fractional B splines using generalized Stirling-type numbers, linking spline theory, fractional calculus, and combinatorics.

Contribution

It derives a new Fourier-type expansion of fractional B splines involving generalized Stirling numbers and introduces a new class of fractional spline polynomials.

Findings

Fourier-type expansion expresses fractional B splines as derivatives of Dirac delta functions.

Explicit shifted distributional representations characterize the action on test functions.

A new class of fractional spline polynomials with generating functions in terms of Mittag Leffler functions.

Abstract

In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function. These results provide a unified framework that connects spline theory, fractional calculus, and combinatorial structures.