Reproducing fractional monomials: weakening of the Strang-Fix conditions

TL;DR

This paper presents a method for reproducing fractional monomials using fractional B-splines, demonstrating a weakening of the Strang-Fix conditions and extending the approach to bivariate cases.

Contribution

It introduces a novel approach to reproduce fractional monomials with integer shifts of fractional B-splines, including bivariate extensions, based on weakened Strang-Fix conditions.

Findings

Reproduces fractional monomials using fractional B-splines.

Extends the method to bivariate fractional B-splines.

Shows the relation between monomial degree and approximation order holds.

Abstract

A method to reproduce causal and symmetric monomials of fractional degree by integer shifts of the corresponding fractional B-splines, introduced by M. Unser and Th. Blue, is presented. Thus the traditional relation between the degree of reproduced monomials and the order of approximation holds. Bivariate, obtained by tensor product, fractional B-splines are introduced; and reproducing of bivariate causal and symmetric monomials is shown. Demonstration that the method is based on a weakening of the Strang-Fix conditions is presented.