Abstract
We consider the problem on optimal quadrature formulas for curvilinear integrals of the first kind that are exact for constant functions. This problem is reduced to the minimization problem for a quadratic form in many variables whose matrix is symmetric and positive definite. We prove that the objective quadratic function attains its minimum at a single point of the corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there exists a unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the least possible norm of the error functional in the conjugate space. We show that the tuple of weights of the optimal quadrature formula is a solution of a special nondegenerate system of linear algebraic equations.
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REFERENCES
S. M. Nikolskiĭ and P. I. Lizorkin, “Approximation by spherical functions,” Trudy Mat. Inst. Steklova 173, 181 (1986) [Proc. Steklov Inst. Math. 173, 195 (1987)].
S. L. Sobolev and V. L. Vaskevich, Cubature Formulas (Izd. Inst. Mat. SO RAN, Novosibirsk, 1996) [The Theory of Cubature Formulas (Kluwer Academic Publishers, Dordrecht, 1997)].
V. L. Vaskevich, “Embedding constants for periodic Sobolev spaces of fractional order,” Sib. Matem. Zh. 49, 1019 (2008) [Siberian Math. J. 49, 806 (2008)].
V. L. Vaskevich, “Embedding constants and embedding functions for Sobolev-like spaces on the unit sphere,” Dokl. Ross. Akad. Nauk 433, 441 (2010) [Dokl. Math. 82, 568 (2010)].
V. L. Vaskevich, “Extremal functions of cubature formulas on a multidimensional sphere and spherical splines,” Mat. Trudy 14, no. 2, 14 (2011) [Siberian Adv. Math. 22, 217 (2012)].
V. L. Vaskevich, “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures,” Sib. Matem. Zh. 53, 1245 (2012) [Siberian Math. J. 53, 996 (2012)].
V. L. Vaskevich, “Spherical cubature formulas in Sobolev spaces,” Sib. Matem. Zh. 58, 530 (2017) [Siberian Math. J. 58, 408 (2017)].
Funding
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. FWNF-2022-0008).
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Vaskevich, V.L., Turgunov, I.M. Optimal Quadrature Formulas for Curvilinear Integrals of the First Kind. Sib. Adv. Math. 34, 80–90 (2024). https://doi.org/10.1134/S1055134424010048
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DOI: https://doi.org/10.1134/S1055134424010048