Multivariate Complex B-Splines, Dirichlet Averages and Difference Operators

For the Schoenberg B-splines, interesting relations between their functional representation, Dirichlet averages and difference operators are known. We use these relations to extend the B-splines to an arbitrary (infinite) sequence of knots and to higher dimensions. A new Fourier domain representation of the multidimensional complex B-spline is given. 1. Complex B-Splines Complex B-splines are a natural extension of the classical Curry-Schoenberg B-splines [2] and the fractional splines first investigated in [16]. The complex B-splines Bz : R→ C are defined in Fourier domain as F(Bz)(ω) = ∫ R Bz(t)e dt = ( 1− e−iω iω )z forRe z > 1. They are well-defined, because of { 1−e −iω iω | ω ∈ R} ∩ {y ∈ R | y < 0} = ∅ they live on the main branch of the complex logarithm. Complex B-splines are elements of L(R) ∩ L(R). They have several interesting basic properties, which are discussed in [5]. Let Re z,Re z1,Re z2 > 1. • Complex B-splines Bz are piecewise polynomials of complex degree. • Smoothness and decay: – Bz ∈ W r 2 (R) for r < Re z − 12 . Here W r 2 (R) denotes the Sobolev space with respect to the L-Norm and with weight (1 + |x|). – Bz(x) = O(x−m) form < Re z+1, |x| → ∞. • Recursion formula: Bz1 ∗Bz2 = Bz1+z2 . • Complex B-splines are scaling functions and generate multiresolution analyses and wavelets. • But in general, they don’t have compact support. • Last but not least: They relate difference and differential operators. In this paper, we take closer look at this last relation and the respective multivariate setting. To this end, we will consider the known relations between classical B-splines, difference operators and Dirichlet averages. B-splines Dirichlet averages Difference operators Figure 1: Relations between classical B-splines, difference operators and Dirichlet averages. 2. Representation in time-domain We defined complex B-splines in Fourier domain, and Fourier inversion shows that these functions are piecewise polynomials of complex degree: Proposition 1. [5] Complex B-splines have a timedomain representation of the form