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 Sibirsk. Mat. Zh., 2012, Volume 53, Number 6, Pages 1245–1262 (Mi smj2379)

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

a Novosibirsk State University, Novosibirsk, Russia
b Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss's quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with $2m^2$ nodes are positive. The formula itself is exact at all spherical harmonics up to order $2m-1$.

Keywords: spherical cubature formula, error functional, Sobolev space on a higher-dimensional sphere, embedding constants and functions, practical error, guaranteed accuracy.

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English version:
Siberian Mathematical Journal, 2012, 53:6, 996–1010

Bibliographic databases:

UDC: 517.518.23+517.518.83+519.651

Citation: V. L. Vaskevich, “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures”, Sibirsk. Mat. Zh., 53:6 (2012), 1245–1262; Siberian Math. J., 53:6 (2012), 996–1010

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. L. Vaskevich, “The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces”, Siberian Math. J., 55:5 (2014), 792–806
2. I. V. Korytov, “Neravenstva Klarksona dlya prostranstva Soboleva periodicheskikh funktsii”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 3, Izd-vo Kazanskogo un-ta, Kazan, 2016, 336–349
3. I. V. Korytov, “Clarkson's inequalities for periodic Sobolev space”, Lobachevskii J. Math., 38:6 (2017), 1146–1155
4. V. L. Vaskevich, “Spherical cubature formulas in Sobolev spaces”, Siberian Math. J., 58:3 (2017), 408–418
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