Fast memory efficient evaluation of spherical polynomials at scattered points

Ivanov, Kamen; Petrushev, Pencho

doi:10.1007/s10444-014-9354-3

Cited by 9 publications

(4 citation statements)

References 23 publications

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“…Either ( 43) and (3), or the needlet decomposition (10) can be used to compute the fully discrete needlet approximation V need J,N (f ). Some discussion of efficient implementation can be found in [16]. We then approximate the L 2 error by a quadrature rule ( w i , x i ) : i = 1, .…”

Section: Needlet Approximation For the Entire Spherementioning

confidence: 99%

Fully discrete needlet approximation on the sphere

Wang

Gia

Sloan

et al. 2017

Applied and Computational Harmonic Analysis

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Spherical needlets are highly localized radial polynomials on the sphere S d ⊂ R d+1 , d ≥ 2, with centers at the nodes of a suitable cubature rule. The original semidiscrete spherical needlet approximation of Narcowich, Petrushev and Ward is not computable, in that the needlet coefficients depend on inner product integrals. In this work we approximate these integrals by a second quadrature rule with an appropriate degree of precision, to construct a fully discrete needlet approximation. We prove that the resulting approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace series partial sum with inner products replaced by appropriate cubature sums. It follows that the L p -error of discrete needlet approximation of order J for 1 ≤ p ≤ ∞ and s > d/p has for a function f in the Sobolev space W s p (S d ) the optimal rate of convergence in the sense of optimal recovery, namely O 2 −Js . Moreover, this is achieved with a filter function that is of smoothness, in contrast to the usually assumed C ∞ . A numerical experiment for a class of functions in known Sobolev smoothness classes gives L 2 errors for the fully discrete needlet approximation that are almost identical to those for the original semidiscrete needlet approximation. Another experiment uses needlets over the whole sphere for the lower levels together with high-level needlets with centers restricted to a local region. The resulting errors are reduced in the local region away from the boundary, indicating that local refinement in special regions is a promising strategy.Proof. By Theorem 3.10, the approximation by the semidiscrete needlets V need J (f ) is equivalent to that by filtered approximation V 2 J−1 ,H (f ). Then the definition (8) of V need J (f ) and (29) of Theorem 3.6 together with Theorem 3.10 giveTheorem 3.11 and Lemma 3.5 imply a rate of convergence of the approximation error of V need J (f ) for f in a Sobolev space, as follows.7 2 (S 2 ). The function f k has limited smoothness at the centers z i and at the boundary of each cap with center z i . These features make f k relatively difficult to approximate in these regions, especially for small k.L 2 approximation error. We show the L 2 errors when using V need J and by V need J,N . For V need J,N (f ) we compute its L 2 error by discretizing the squared L 2 -norm by a quadrature rule. We cannot compute

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Section: Needlet Approximation For the Entire Spherementioning

confidence: 99%

Fully discrete needlet approximation on the sphere

Wang

Gia

Sloan

et al. 2017

Applied and Computational Harmonic Analysis

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“…Chapter 5 Fully discrete needlet approximations on the sphere 103 as named by Sloan and Womersley [68]; see also [41] and [35].…”

Section: 3)mentioning

confidence: 99%

“…Either (5.3.10) and (2.6.5), or the needlet decomposition (5.1.6) can be used to compute the fully discrete needlet approximation V need L,N (f ). Some discussion of efficient implementation can be found in [35]. We then approximate the L 2 error by a quadrature rule ( w i , x i ) : i = 1, .…”

Section: 4 Numerical Examplesmentioning

confidence: 99%

On Filtered Polynomial Approximation on the Sphere

Wang

Sloan

2016

J Fourier Anal Appl

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Localised polynomial approximations on the sphere have a variety of applications in areas such as signal processing, geomathematics and cosmology. Filtering is a simple and effective way of constructing a localised polynomial approximation. In this thesis we investigate the localisation properties of filtered polynomial approximations on the sphere. Using filtered polynomial kernels and a special numerical integration (quadrature) rule we construct a fully discrete need let approximation. The localisation of the filtered approximation can be seen from the localisation properties of its convolution kernel. We investigate the localisation of the filtered Jacobi kernel, which includes the convolution kernel for filtered approximation on the sphere as a particular example. We prove the precise relation between the filter smoothness and the decay rate of the corresponding filte red Jacobi kernel over local and global regions. The difference in localisation properties between Fourier and filtered approximations can be illustrated by their Riemann localisation. We show that the Riemann localisation property holds for the Fourier-Laplace partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimensions. We then prove that the filtered approximation with sufficiently smooth filter has the Riemann localisation property for spheres of any dimensions. Filtered convolution kernels with a special filter become spherical need lets, which are highly localised zonal polynomials on the sphere with centres at the nodes of a suitable quadrature rule. The original semidiscrete spherical needlet approximation has coefficients defined by inner product integrals. We use an appropriate quadrature rule to construct a fully discrete version. We prove that the fully discrete spherical need let approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace partial sum with inner products replaced by appropriate quadrature sums. From this we establish error bounds for the fully discrete need let approximation of functions in Sobolev spaces on the sphere. The power of the need let approximation for local approximation is shown by numerical experiments that use low-level need lets globally together with high-level need lets in a local region. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media , now or here after known , subject to the provisions of the Copyright Act 1968. I retain all property rights , such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation.

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“…The kernel is highly localized if it decays at rates faster than any inverse polynomial rate away from the main diagonal y = x in Ω × Ω with respect to the distance d on Ω; see the definition in the next section. These kernels provide important tools for analysis on regular domains, such as the unit sphere and the unit ball, and are essential ingredient in recent study of approximation and localized polynomial frames; see, for example, [3,8,11,14,17,18,19] for some of the results on the spheres and balls and [1,2,12,13,15,25] for various applications. The reason that highly localized kernels are known only on a few regular domains lies in the addition formula for orthogonal polynomials, which are closed form formulas for the reproducing kernels of orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

Approximation and localized polynomial frame on double hyperbolic and conic domains

Xu¹

2021

Preprint

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We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in [30] for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed form formulas for the reproducing kernels. The main results provide a construction of semi-discrete localized tight frame in weighted L 2 norm and a characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

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Fast memory efficient evaluation of spherical polynomials at scattered points

Cited by 9 publications

References 23 publications

Fully discrete needlet approximation on the sphere

Fully discrete needlet approximation on the sphere

On Filtered Polynomial Approximation on the Sphere

Approximation and localized polynomial frame on double hyperbolic and conic domains

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