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

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 rectangular lattice Q_k,i = { x_k-1 ≤ x ≤ x_k , y_i-1 ≤ y ≤ y_i} are not included in the number of nodes cubature formula

Z Z (Q) f(x, y)dxdy = Xm k=1 Xn i=1 pkif(xk, yi) + Rmn(f), (1)

the formula of average rectangles is the best for classes of functions ^ω1,ω2 (Q), H^ω1_p1 (Q) and H^ω1_p2(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.