Statistical Encounters with Complex B-Splines

Journal of Approximation Theory

2010

Self Cite

The notion of a complex B-spline is extended to a multivariate setting by means of ridge functions employing the known geometric relationship between ordinary B-splines and multivariate B-splines. To derive properties of complex B-splines in R s , 1 < s ∈ N, the Dirichlet average has to be generalized to include infinite-dimensional simplices ∆ ∞ . Based on this generalization several identities of multivariate complex B-splines are presented.

“…The existence of µ b is also discussed in [13,26,14,15]. For a detailed discussion of the measure µ b and the simplex ∆ ∞ , we also refer to [9] and [8].…”

Section: Dirichlet Averagesmentioning

confidence: 99%

“…The infinite sum exists whenever lim sup n→∞ n τ n ≤ , some ∈ [0, e), (4.2) where · now denotes the canonical Euclidean norm on C s . (See also [8]. )…”

Section: Dirichlet Averagesmentioning

confidence: 99%

See 1 more Smart Citation

Multivariate complex B-splines and Dirichlet averages

Journal of Approximation Theory

2010

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“…Furthermore, complex B-splines relate in a natural fashion to difference and differential operators of fractional order. For further properties and relationships to Dirichlet averages and fractional derivatives and integrals we refer the interested reader to [4,5].…”

Section: Introductionmentioning

confidence: 99%

Complex B-splines and Hurwitz zeta functions

Garunk

et al. 2013

LMS J. Comput. Math.

“…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]. …”

mentioning

confidence: 99%

Multivariate Interpolation with Fundamental Splines of Fractional Order

Proc Appl Math and Mech

2011

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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,