Optimal quadratures in the sense of sard in a Hilbert space. (English) Zbl 1390.41038
Summary: 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 \(\sin \omega x, \cos \omega x\), 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.
MSC:
| 41A55 | Approximate quadratures |
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
| 41A15 | Spline approximation |
| 41A50 | Best approximation, Chebyshev systems |
Keywords:
optimal quadrature formulas; error functional; extremal function; Hilbert space; optimal coefficientsReferences:
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