Optimal quadrature problem on classes defined by kernels satisfying certain oscillation properties

Abstract

We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let $\lfloor{x}\rfloor$ be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula

$$Q^*_N(f) = \frac{2\pi}{N}\sum_{j=0}^{N-1} f\left(\frac{2\pi j}{N}\right)$$

is optimal among all quadrature formulae of the form

$$Q_{2N}(f) = \sum_{i=1}^{n}\sum_{j=0}^{\nu_{i}-1}a_{ij}f^{(j)}(t_{i}),$$

where the nodes 0 ≤  t ₁ < ... < t _n  < 2π and the coefficients (weights) $a_{ij}\in \mathbb{R}$ are arbitrary, i = 1,...,n, j = 0,1,..., ν _i − 1, and (ν₁,...,ν _n ) is a system of positive integers satisfying the condition $\mathop{\sum}_{i=1}^{n}2\lfloor{(\nu_i+1)/2}\rfloor\leq 2N$. In particular, the rectangular formula is optimal for these classes of functions among all quadrature formulae of the form

$$Q_N(f) = \sum_{i=1}^{N}a_{i}f(t_{i}),$$

with free nodes 0 ≤  t ₁ <  ... < t _N <  2π and arbitrary weights $a_{i}\in \mathbb{R}, i=1,\ldots,N$. Moreover, we exactly determine the error estimates of the optimal quadrature formulae on these classes of functions.