Inner products of box splines and their derivatives

Speleers, Hendrik

doi:10.1007/s10543-014-0513-1

Cited by 8 publications

(4 citation statements)

References 18 publications

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“…The inner product formula for cardinal B-splines traces back to[44]. The formula for derivatives of cardinal B-splines can be found in[21] and a generalization for multivariate box splines in[48].…”

mentioning

confidence: 99%

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

Lyche

Manni

Speleers

2018

Lecture Notes in Mathematics

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This chapter presents an overview of polynomial spline theory, with special emphasis on the B-spline representation, spline approximation properties, and hierarchical spline refinement. We start with the definition of B-splines by means of a recurrence relation, and derive several of their most important properties. In particular, we analyze the piecewise polynomial space they span. Then, we present the construction of a suitable spline quasi-interpolant based on local integrals, in order to show how well any function and its derivatives can be approximated in a given spline space. Finally, we provide a unified treatment of recent results on hierarchical splines. We especially focus on the so-called truncated hierarchical B-splines and their main properties. Our presentation is mainly confined to the univariate spline setting, but we also briefly address the multivariate setting via the tensor-product construction and the multivariate extension of the hierarchical approach.

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confidence: 99%

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

Lyche

Manni

Speleers

2018

Lecture Notes in Mathematics

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“…Remark Theorem 3 is a generalization of a known explicit formula for inner products of integer derivatives of cardinal B‐splines (see Reference 42 and also References 39 and 43). …”

Section: Fractional Derivatives Of Cardinal B‐splinesmentioning

confidence: 99%

On the matrices in B‐spline collocation methods for Riesz fractional equations and their spectral properties

Mazza

Donatelli

Manni

et al. 2022

Numerical Linear Algebra App

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In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol vanishes at 0 with order 𝛼, the fractional derivative order that ranges from 1 to 2, and it decays exponentially to zero at 𝜋 for increasing p at a rate that becomes faster as 𝛼 approaches 1. This translates into a mitigated conditioning in the low frequencies and into a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with nonfractional diffusion problems, the approximation order for smooth solutions in the fractional case is p + 2 − 𝛼 for even p, and p + 1 − 𝛼 for odd p.

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“…Remark 3.4. Theorem 3.3 is a generalization of a known explicit formula for inner products of integer derivatives of cardinal B-splines (see [15,23] and also [36]).…”

Section: Fractional Derivatives Of Cardinal B-splinesmentioning

confidence: 99%

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

Mazza¹,

Donatelli²,

Manni³

et al. 2021

Preprint

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In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol has a single zero at 0 of order α, with α the fractional derivative order that ranges from 1 to 2, and it presents an exponential decay to zero at π for increasing p that becomes faster as α approaches 1. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is p + 2 − α for even p, and p + 1 − α for odd p.

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Inner products of box splines and their derivatives

Cited by 8 publications

References 18 publications

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

On the matrices in B‐spline collocation methods for Riesz fractional equations and their spectral properties

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

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