A combination of downward continuation and local approximation for harmonic potentials

Gerhards, Christian

doi:10.1088/0266-5611/30/8/085004

Cited by 16 publications

(16 citation statements)

References 51 publications

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“…[2,5,23,25,52,65]. The Riemann localisation property of the Fourier-Laplace partial sum for W s p (S 2 ) implies that the multiscale approximation converges to the solution of the local downward continuation problem, see [23,30]. The estimation of the Fourier local convolution also plays a role in the "missing observation" problem, see [44,Section 10.5] and [6].…”

Section: Fourier Casementioning

confidence: 99%

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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: Fourier Casementioning

confidence: 99%

“…This effect, established in Lemma 4.2.5, is one of the vital factors in the proof of the Riemann localisation property. It is interesting to note that the result for the operator norm is not strong enough for application to the local downward continuation problem [30] with d = 2.…”

Section: Fourier Casementioning

confidence: 99%

On Filtered Polynomial Approximation on the Sphere

Wang

Sloan

2016

J Fourier Anal Appl

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“…[1,2,12,13,19,20,26,34]. The Riemann localisation property of the Fourier-Laplace series partial sum for W s p (S 2 ) implies that the multiscale approximation converges to the solution of the local downward continuation problem, see [12,15]. The estimation of the local convolution also plays a role in the "missing observation" problem, see [18,Section 10.5] and [3].…”

Section: Filtered Casementioning

confidence: 99%

Riemann Localisation on the Sphere

Wang

Sloan

Womersley

2016

J Fourier Anal Appl

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This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the twodimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere S d ⊂ R d+1 , d ≥ 2, we mean that for a suitable subsetof the Fourier local convolution of f ∈ X converges to zero as the degree goes to infinity. The Fourier local convolution of f at x ∈ S d is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of x. The failure of Riemann localisation for d > 2 can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.

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“…A frequently used choice of basis functions for the spectral method are the spherical harmonics [23]. They are eigenfunctions of the spherical Laplacian, which makes them a natural choice for problems such as solving differential equations [39], spherical deconvolution [16], the approximation of measures [11], or modeling the earth's upper mantle [15,38] and gravitational field [13]. While the naive computation of spherical harmonics expansions is very memory and time consuming [39], there are methods allowing for a faster computation, known as fast spherical Fourier transforms, see, e.g., [10,19,24,37] and [28, sec.…”

Section: Introductionmentioning

confidence: 99%

Approximation properties of the double Fourier sphere method

Mildenberger,

Quellmalz

2021

Preprint

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We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical Hölder spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence.

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A combination of downward continuation and local approximation for harmonic potentials

Cited by 16 publications

References 51 publications

On Filtered Polynomial Approximation on the Sphere

On Filtered Polynomial Approximation on the Sphere

Riemann Localisation on the Sphere

Approximation properties of the double Fourier sphere method

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