Fractional B-splines Bσ, σ ≥ 1, are piecewise polynomials of fractional degree that interpolate the classical Schoenberg splines Bn, n ∈ N, with respect to the degree. As the Schoenberg splines of order ≥ 3, they in general do not satisfy the interpolation property Bσ(n−k) = δ n,k , n, k ∈ Z. However, the application of the interpolation filter 1/ k∈Z Bσ(ω−2πk)-if well-defined-in the frequency domain yields a fundamental spline of fractional order that does satisfy the interpolation property. We extend these result via ridge functions to multivariate fractional B-splines.
Univariate fractional B-SplinesFractional B-splines, which were first investigated in [5], are a natural extension of the classical Curry-Schoenberg (polynomial) B-splines. These fractional B-splines B σ : R → C are defined in the Fourier domain by. Then, since graph Θ ∩ {y ∈ R | y < 0} = ∅, fractional B-splines reside on the main branch of the complex logarithm and are thus well-defined. Moreover, they are elements of L 1 (R) ∩ L 2 (R) and possess several interesting basic properties, which are discussed in [5]. E.g. they satisfy smoothness and decay conditions depending on σ; they are scaling functions, and generate multiresolution analyses and wavelets. But in general, fractional B-splines do not have compact support. Furthermore, they relate difference and differential operators. For the relationships to Dirichlet averages, fractional derivatives and integrals, and for results concerning multivariate fractional B-splines we refer to [1,3].
Multivariate fractional B-splinesIn this article, we restrict ourselves to a special class of multivariate B-splines . Let the sequence of knots τ of the multivariate splines be equidistantly distributed along a ray in R s , s ∈ N, i.e., let τ = dN 0 for some distance vector d ∈ R s . Then we define the multivariate B-spline B σ (x | τ ) for x ∈ R s in the Fourier domain via ridge functions and the univariate B-spline. To this end, let ω, λ ∈ R s , ω = ω . We defineDue to the construction via ridges,is bounded in a neighborhood of the origin and decays as ω −σ as ω → ∞, the function is also an element of L 1 (R s ), as long as σ > s. For more details and images of these splines we refer to [2].
The interpolation problemOur goal is to construct a multivariate fundamental cardinal spline of fractional order L σ : R s → C,for an appropriate bi-infinite sequence {c : k ∈ Z s }, a formula for L σ is obtained:. (3) * Brigitte Forster: email forster@ma.tum.de,