On Asymptotic Properties of Certain $B$-Splines in Terms of Theta-like Functions

TL;DR

This paper investigates the asymptotic behavior of Mellin transforms of specific $B$-splines with special knots, revealing their relation to theta-like functions and providing pointwise asymptotics.

Contribution

It introduces new asymptotic formulas for $B$-splines' Mellin transforms using theta-like functions and polynomial interpolation techniques.

Findings

Asymptotic formulas for Mellin transforms of $B$-splines derived.

Pointwise asymptotics of $B_N^*$ and $B_N$ discussed.

Connection established between $B$-splines and theta-like functions.

Abstract

The asymptotic behavior of the Mellin transform of the associated $B$-splines $B_N^*(t) :=t^{-N}B_N(t)$ with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and properties of certain theta-like functions. Pointwise asymptotics of $B_N^*$ and $B_N$ are discussed as well.