Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.15244

Fourier representations of fractional B Splines via generalized Stirling type polynomials

Damla Gun·Peter Massopust·Yilmaz Simsek
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Damla Gun, Peter Massopust, Yilmaz Simsek. 2026-05-14. Fourier representations of fractional B Splines via generalized Stirling type polynomials. https://arxiv.org/abs/2605.15244

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗