Моносплайны и наилуч шие квадратурные фор мулы для некоторых классо в непериодических фу нкций

Abstract

The following quadrature formulae are considered: $$\int\limits_0^1 {f(x)dx = \mathop \sum \limits_{k = 1}^n a_k f(x_k ) + \mathop \sum \limits_{i = 1}^l } b_i f^{(\alpha _i )} (0) + \mathop \sum \limits_{j = 1}^m c_j f^{(\beta _j )} (1) + R(f),$$ where 0≦x1<x2<...<xn≦1 0≦α i , βj≦r−1;l, m, n, andr are positive integers. The problem of existence and uniqueness of the best quadrature formula is solved for the classesW r L p (r=1, 2, ...; 1<p≦∞), obtaining the characteristic properties of its nodesx k and weightsa k ,b i , andc j .