Abstract
We study sequences of cubature formulas on the unit sphere in a multidimensional Euclidean space. The grids for the cubature formulas under consideration embed in each other consecutively, forming in the limit a dense subset on the initial sphere. As the domain of cubature formulas, i.e. as the class of integrands, we take spherical Sobolev spaces. These spaces may have fractional smoothness. We prove that, among all possible spherical cubature formulas with given grid, there exists and is unique a formula with the least norm of the error, an optimal formula. The weights of the optimal cubature formula are shown to be solutions to a special nondegenerate system of linear equations. We prove that the errors of cubature formulas tend to zero as the number of nodes grows indefinitely.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sobolev S. L. and Vaskevich V. L., The Theory of Cubature Formulas, Kluwer Academic Publishers, Dordrecht (1997).
Vaskevich V. L., “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures,” Sib. Math. J., vol. 53, no. 6, 996–1010 (2012).
Triebel H., “Sampling numbers and embedding constants,” Proc. Steklov Inst. Math., vol. 248, 275–284 (2005).
Vaskevich V. L., “Embedding constants and embedding functions for Sobolev-like spaces on the unit sphere,” Dokl. Math., vol. 82, no. 1, 568–572 (2010).
Vaskevich V. L., “Embedding constants for periodic Sobolev spaces of fractional order,” Sib. Math. J., vol. 49, no. 5, 806–813 (2008).
Vaskevich V. L., “Extremal functions of cubature formulas on a multidimensional sphere and spherical splines,” Siberian Adv. Math., vol. 22, no. 3, 217–226 (2012).
Ignatov M. I. and Pevnyi A. B., Natural Splines in Several Variables [Russian], Nauka, Leningrad (1991).
Mikhlin S. G., Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford, London, Edinburgh, New York, Paris, and Frankfurt (1965).
Müller C., Spherical Harmonics, Springer-Verlag, Berlin (1966).
Sobolev S. L., Introduction to the Theory of Cubature Formulas, Gordon and Breach Science Publishers, Montreux (1974).
Lizorkin P. I. and Nikol’skii S. M., “Approximation by spherical functions,” Proc. Steklov Inst. Math., vol. 173, 195–203 (1987).
Salikhov G. N., Cubature Formulas for Multidimensional Spheres [Russian], Fan, Tashkent (1985).
Yosida K., Functional Analysis, Springer-Verlag, Berlin (1994).
Holmes R., A Course on Optimization and Best Approximation, Springer-Verlag, Berlin, Heidelberg, and New York (1972) (Lect. Notes Math.; vol. 257).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2017 Vaskevich V.L.
Rights and permissions
About this article
Cite this article
Vaskevich, V.L. Spherical cubature formulas in Sobolev spaces. Sib Math J 58, 408–418 (2017). https://doi.org/10.1134/S0037446617030053
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446617030053