General interpolation on the lattice $h{\Bbb Z}^s$ : Compactly supported fundamental solutions

Riemenschneider, S. D.; Shen, Zuowei

doi:10.1007/s002110050123

Cited by 8 publications

(3 citation statements)

References 13 publications

(34 reference statements)

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“…The unisolvency of the local interpolation problem in Section 5.1 poses a challenging problem for further study. This problem is related to the existence of fundamental functions with compact support for cardinal interpolation in cardinal spline spaces, [14,15], see also [1]. Multivariate Euler-Frobenius polynomials [4,11] seem to be promising tools to face it.…”

Section: Resultsmentioning

confidence: 99%

Quasi-interpolation projectors for box splines

Lyche¹,

Manni²,

Sablonnière³

2008

Journal of Computational and Applied Mathematics

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Section: Resultsmentioning

confidence: 99%

Quasi-interpolation projectors for box splines

Lyche¹,

Manni²,

Sablonnière³

2008

Journal of Computational and Applied Mathematics

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“…Qi's result has been extended by Dahmen et al [7] to splines with nonuniform knots. A multidimensional analogue of Qi's results can be found in [25].…”

Section: For Any Integermentioning

confidence: 89%

Spline Interpolation and Wavelet Construction

Lee

Sharma

Tan

1998

Applied and Computational Harmonic Analysis

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The method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages from B-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of ''good'' filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression.᭧ 1998 Academic Press

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“…Previously, only cardinal box spline interpolation (data on a regular grid) has been considered (see [BHR93] and [RS95] for fundamental solutions which are of exponential decay and compact support, respectively). The multilevel scheme proposed here is based on a generalisation of the least solution of the polynomial interpolation problem of de Boor and Ron [BR92] to an arbitrary inner product space of interpolants by Waldron [W99], and uses the box spline wavelet decomposition of Riemenschneider and Shen [RS92].…”

Section: Introductionmentioning

confidence: 99%

Scattered data interpolation by box splines

Lau¹,

Xin²,

Yau³

2008

AMS/IP Studies in Advanced Mathematics

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Given scattered data in IR s , interpolation from a dilated box spline space S M (2 k ·)is always possible for a fine enough scaling. For example, for the Lagrange function of a point θ one could take any shifted dilate M (2 k · −j) which is nonzero at θ and zero at the other interpolation points. However, the resulting interpolant, though smooth (and local), will consist of a set of "bumps", and so by any reasonable measure provides a poor representation of the shape of the underlying function. On the other hand, it is possible to choose a space of interpolants which contains some M (2 k · −j) of arbitrarily large support. But the resulting methods are increasingly less local, and in general still require some splines with a much higher level of dilation. Here we provide a multilevel method which constructs a space of interpolants by taking as many splines as possible from a given dilation level, then as many from the next (higher) dilation level, and so forth. The choice at each level is made using the suggestion of [W99], which is based on the Riesz representation theorem. This requires an inner product on the ground space S M , and the higher levels. .. The inner products used here involve the box spline coefficients, and prewavelet coefficients of [RS92], respectively, and are norm equivalent to · L 2 (IR s ) . These lead to a scheme which is easily implemented, and numerically stable. Previously, box spline interpolants have been considered only for data on a regular grid.

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General interpolation on the lattice $h{\Bbb Z}^s$ : Compactly supported fundamental solutions

Cited by 8 publications

References 13 publications

Quasi-interpolation projectors for box splines

Quasi-interpolation projectors for box splines

Spline Interpolation and Wavelet Construction

Scattered data interpolation by box splines

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