Fourier-Laplace series on a sphere

Zhizhiashvili, Levan; Topuriya, S. B.

doi:10.1007/bf01844489

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Introduction3

The Integral Operators1

Generalization To Multidimensions1

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2018

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Cited by 12 publications

(5 citation statements)

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“…Il'in and E.M. Nikishin in [3]. The summability problems of Fourier-Laplace series on the sphere described in the book [15] by L.V. Zhizhiashvili and S.B.…”

Section: The Integral Operatorsmentioning

confidence: 99%

Summation by Riesz means of the Fourier-Laplace series

Rasedee

Rakhimov

Akhmedov

2018

Filomat

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“…Il'in and E.M. Nikishin in [3]. The summability problems of Fourier-Laplace series on the sphere described in the book [15] by L.V. Zhizhiashvili and S.B.…”

Section: The Integral Operatorsmentioning

confidence: 99%

Summation by Riesz means of the Fourier-Laplace series

Rasedee

Rakhimov

Akhmedov

2018

Filomat

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“…where R −1 is defined further in (4.14). Under assumptions (4.2), (4.4), for each fixed x, series of (4.5) converges in L 2 (S n−1 ); see, e.g., [SW16] (Chapter 4), [Mor98] (Chapter 2), [ZT79].…”

Section: Generalization To Multidimensionsmentioning

confidence: 99%

“…(1.11) see also [SW16], [ZT79] for other properties of the associated Legendre polynomials. In addition, coefficients w k,n in (1.7) are defined by the formulas:…”

Section: Introductionmentioning

confidence: 99%

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An iterative inversion of weighted Radon transforms along hyperplanes

Goncharov¹

2017

Inverse Problems

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We propose iterative inversion algorithms for weighted Radon transforms RW along hyperplanes in R 3 . More precisely, expanding the weight W = W (x, θ), x ∈ R 3 , θ ∈ S 2 , into the series of spherical harmonics in θ and assuming that the zero order term w0,0(x) = 0, x ∈ R 3 , we reduce the inversion of RW to solving a linear integral equation. In addition, under the assumption that the even part of W in θ (i.e., 1 2 (W (x, θ) + W (x, −θ))) is close to w0,0, the aforementioned linear integral equation can be solved by the method of successive approximations. Approximate inversions of RW are also given. Our results can be considered as an extension to 3D of two-dimensional results of Kunyansky (1992), Novikov (2014, Guillement, Novikov (2014). In our studies we are motivated, in particular, by problems of emission tomographies in 3D. In addition, we generalize our results to the case of dimension n > 3.

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“…In the case of Spectral expansions of the integrable functions on unit sphere the same problems are considered in the papers [1], [2], [3], [6], [8], [12] and [15] . For more reference related to the problems of the convergence of the Fourier-Laplace series on unit sphere we refer the readers to [20]. Investigating the problems of the localization of the FourierLaplace series of the distribution on unit sphere was started by Rakhimov (see [16]).…”

Section: Introductionmentioning

confidence: 99%