Symbols and exact regularity of symmetric pseudo-splines of any arity

TL;DR

This paper derives explicit formulas for the symbols of symmetric pseudo-spline subdivision schemes of any arity, demonstrating their regularity and positivity properties through algebraic and spectral analysis.

Contribution

It introduces a generating function approach to compute exact regularity of symmetric pseudo-splines of any arity, extending previous results to a broader class.

Findings

Symbols have positive Fourier transform.

Regularity can be computed algebraically via spectral radius.

Explicit regularity formulas for binary, ternary, and quaternary cases.

Abstract

Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc-Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric $m$-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact H\"older regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes.