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.
Key words:quadrature formula, error functional, Sobolev space on a closed curve, embedding constant and function, optimal formula.
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