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

Michael I. Ganzburg

TL;DR

The paper addresses the asymptotics of associated B-splines $B_N^*(t)=t^{-N}B_N(t)$ through Mellin transforms and theta-like functions, building on polynomial interpolation of power functions. By connecting interpolation errors to Mellin integrals and introducing theta-like functions $\Theta_d$, it proves limit relations $\lim_{N\to\infty} A_N M(B_N^*(C_N\cdot,\Omega),s)=M(\Theta(\cdot),s)$ and, under a $(r,\beta)$-Condition, pointwise limits $\lim_{N\to\infty} A_N B_N^*(C_N t,\Omega)=\Theta_d(t)$ for selected knot sets. The work unifies Gaussian-type limits with non-Gaussian theta-like limits, providing explicit asymptotic formulas for various knot families (including Chebyshev and equidistant-zero knots) and offering a framework for delta-convergence analyses. These results extend classical spline asymptotics and reveal theta-analytic structures in spline limits, with potential implications for approximation theory and numerical analysis of spline-based kernels.

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.

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

TL;DR

The paper addresses the asymptotics of associated B-splines through Mellin transforms and theta-like functions, building on polynomial interpolation of power functions. By connecting interpolation errors to Mellin integrals and introducing theta-like functions , it proves limit relations and, under a -Condition, pointwise limits for selected knot sets. The work unifies Gaussian-type limits with non-Gaussian theta-like limits, providing explicit asymptotic formulas for various knot families (including Chebyshev and equidistant-zero knots) and offering a framework for delta-convergence analyses. These results extend classical spline asymptotics and reveal theta-analytic structures in spline limits, with potential implications for approximation theory and numerical analysis of spline-based kernels.

Abstract

The asymptotic behavior of the Mellin transform of the associated -splines 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 and are discussed as well.
Paper Structure (9 sections, 19 theorems, 106 equations)

This paper contains 9 sections, 19 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Asymptotics for $B$-splines
  3. Notation and Preliminaries
  4. More Preliminaries: Special Sequences of Polynomials
  5. Polynomial Interpolation of Power Functions
  6. Properties of $B$-splines and Associated $B$-splines
  7. Theta-like Functions and their Properties
  8. The Mellin Transform
  9. Asymptotics of $B_N^*$

Key Result

Theorem 1.1

A monotone function $F:{\mathbb R}\to [0,{\infty})$, with $F(-{\infty})=0$ and $F({\infty})=1$, is a limit of $B$-spline distributions $F_N(x):=\int_{-{\infty}}^x B_N\left(t,{\Omega}\right)dt$ as $N\to{\infty}$ for all x at which $F$ is continuous if and only if $F(x)=\int_{-{\infty}}^x \Lambda(t)dt

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Definition 1.7
  • Example 1.8
  • Example 1.9
  • Example 1.10
  • ...and 43 more