On generalized hyperinterpolation on the sphere

Feng, Dagan

doi:10.1090/s0002-9939-06-08421-8

Cited by 28 publications

(15 citation statements)

References 12 publications

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“…A central assumption in [45,47], in addition to polynomial exactness, is that a Marcinkiewicz-Zygmund (M-Z) inequality is satisfied. Quadrature rules with positive weights and polynomial exactness automatically satisfy an M-Z inequality (see Dai [18,Theorem 2.1] and Mhaskar [47,Theorem 3.3]). However, neither decomposition of wavelets into needlets nor numerical implementation were studied in [45,47].…”

Section: Norms Of Fourier Local Convolutions and Their Kernelsmentioning

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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Section: Norms Of Fourier Local Convolutions and Their Kernelsmentioning

confidence: 99%

On Filtered Polynomial Approximation on the Sphere

Wang

Sloan

2016

J Fourier Anal Appl

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“…for s ∈ N is the quadrature weights of the quadrature rule Q Λ ,s = {(w i,s , x i ) : w i,s ≥ 0 and x i ∈ Λ } with 0 ≤ w i,s ≤ c 1 |D| −1 . It is easy to see that the WRLS estimator in (10) is a batch version of DWRLS, which assumes that all data are stored on a single large server and WRLS is capable of handling them. According to Lemma 2, since the matrix-inversion is involved in WRLS, it requires O(|D| 2 ) memory requirements and O(|D| 3 ) float-computations to solve the optimization problem in (10), which is infeasible when the data size is huge even if all the data could be collected without considering the data privacy issue.…”

Section: Approximation Capability Of Wrlsmentioning

confidence: 99%

“…It is easy to see that the WRLS estimator in (10) is a batch version of DWRLS, which assumes that all data are stored on a single large server and WRLS is capable of handling them. According to Lemma 2, since the matrix-inversion is involved in WRLS, it requires O(|D| 2 ) memory requirements and O(|D| 3 ) float-computations to solve the optimization problem in (10), which is infeasible when the data size is huge even if all the data could be collected without considering the data privacy issue. The study of approximation capability of WRLS (10) is necessary, since it enhances the understanding of DWRLS by means of determining which conditions are sufficient to guarantee that distributed learning performs similarly to its batch counterpart.…”

Section: Approximation Capability Of Wrlsmentioning

confidence: 99%

“…Therefore, we have φk ≤ ψk and consequently N φ ⊆ N ψ . Our following theorem whose proof will be given in Section 5 presents an estimate for the approximation error of WRLS (10) under the metric of N ψ .…”

Section: Approximation Capability Of Wrlsmentioning

confidence: 99%

“…To prove Proposition 1, we need the following spherical Marcinkiewicz-Zygmund inequalities for spherical polynomials [9,10].…”

Section: Spherical Quadrature Rulesmentioning

confidence: 99%

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Radial Basis Function Approximation with Distributively Stored Data on Spheres

Han¹,

Lin²

2021

Preprint

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This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) based on spherical radial basis functions and spherical quadrature rules to tackle spherical data that are stored across numerous local servers and cannot be shared with each other. Via developing a novel integral operator approach, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm with the whole data on a large enough machine. This interesting finding implies that distributed learning is capable of sufficiently exploiting potential values of distributively stored spherical data, even though every local server cannot access all the data.

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