TY - JOUR

T1 - Optimal Quadrature Formulas for Curvilinear Integrals of the First Kind

AU - Vaskevich, V. L.

AU - Turgunov, I. M.

N1 - The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. FWNF-2022-0008).

PY - 2024/3

Y1 - 2024/3

N2 - We consider the problem on optimal quadrature formulas for curvilinear integrals ofthe first kind that are exact for constant functions. This problem is reduced to the minimizationproblem 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 ofthe corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there existsa unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the leastpossible norm of the error functional in the conjugate space. We show that the tuple of weights ofthe optimal quadrature formula is a solution of a special nondegenerate system of linear algebraicequations.

AB - We consider the problem on optimal quadrature formulas for curvilinear integrals ofthe first kind that are exact for constant functions. This problem is reduced to the minimizationproblem 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 ofthe corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there existsa unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the leastpossible norm of the error functional in the conjugate space. We show that the tuple of weights ofthe optimal quadrature formula is a solution of a special nondegenerate system of linear algebraicequations.

KW - Sobolev space on a closed curve

KW - embedding constant and function

KW - error functional

KW - optimal formula

KW - quadrature formula

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85187443320&origin=inward&txGid=4340c92f7939e89d650228d2ca068451

UR - https://www.elibrary.ru/item.asp?id=67307189

UR - https://www.mendeley.com/catalogue/206e4add-38c5-3b40-91b5-5a44f4a2b428/

U2 - 10.1134/S1055134424010048

DO - 10.1134/S1055134424010048

M3 - Article

VL - 34

SP - 80

EP - 90

JO - Siberian Advances in Mathematics

JF - Siberian Advances in Mathematics

SN - 1055-1344

IS - 1

M1 - 4

ER -