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