Optimal quadratures in the sense of Sard in a Hilbert space

An optimal quadrature formula in the sense of Sard in the Hilbert space K 2 ( P m ) is constructed. New optimal quadrature formula of such a type and explicit expressions for the corresponding optimal coefficients are obtained using S.L. Sobolev's method. The obtained optimal quadrature formula is exact for the trigonometric functions sin ω x , cos ω x , and for algebraic polynomials of degree m - 3 . Finally, some numerical results for the norm of the error functional of the optimal quadrature formulas are presented.