On an Optimal Quadrature Formula for Classes of Functions Given by Modulus of Continuity

M. Sh. Shabozov

Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 3, pp. 91-105

Résumé

The problem of minimizing the error of a cubature formula on the classes of functions given by modulus of continuity for cubature formulas with fixed nodes on the boundary of gird rectangular localization domain of nodes is considered. We give the exact solution of this problem on the wide classes of functions of two variables. It was previously shown by N.P. Korneychuk that if the boundary nodes of a \linebreak rectangular lattice $Q_{ki}=\{\, x_{k-1}\le x\le x_{k},\, y_{i-1}\le y\le y_{i}\}$ are not included in the number of nodes cubature formula $$ \iint\limits_{(Q)}f(x,y)dxdy=\sum_{k=1}^m\sum_{i=1}^n p_{ki}f(x_k,y_i)+R_{mn}(f),\qquad\qquad\qquad\qquad\qquad\qquad\qquad(1) $$ the formula of average rectangles is the best for classes of functions $H^{\omega_{1},\omega_{2}}(Q),$  $H_{\rho_{1}}^{\omega}(Q)$  and  $H_{\rho_{2}}^{\omega}(Q)$ among all quadrature formulas of the form (1). It is proved that if into the number of nodes in the formula (1) all boundary nodes (such formulas are called Markov-type) are added, then for these classes of functions the best formula is trapezoids. The exact errors for all classes of functions are calculated.

Détail

Keywords: optimal formulas, extremal problem, formula of Markov type, modulus of continuity, error estimate.
Mots-clés : notes and coefficients