Optimal quadrature formulae for differentiable functions

Abstract

Let Ly:=y(r)+ar-1(x)·y(r-1)+...+ao(x)y be a given linear differential operator which has the property W of Polya in [a,b]. We prove the existence and characterize the optimal quadrature formula of the form $$\int\limits_a^b {f(x)dx} \approx \mathop \sum \limits_{k = 1}^n \mathop \sum \limits_{\lambda = 0}^{v_k - 1} a_{k\lambda } f^{(\lambda )} x_k $$ (with preassigned multiplicities (v k)1n) in the classes $$LW_q^r [a,b]: = \{ f \in C^{(r - 1)} [a,b]:f^{(r - 1)} - abs. cont., \left\| {Lf} \right\|_q \leqslant 1\} $$ for 1<q≤∞.