On One Problem of Optimization of Approximate Integration

TL;DR

This paper proves the optimality of a specific type of interval quadrature formula with equal weights and equidistant nodes for approximating integrals of convolutions of certain kernels with functions in the unit ball of L1, within a particular function class.

Contribution

It establishes the optimality of a class of interval quadrature formulas with equal weights and equidistant nodes for a specific convolution class, extending understanding of approximate integration methods.

Findings

The quadrature formula is proven optimal within the specified class.

Equal weights and equidistant nodes are optimal for the given problem.

The result applies to convolutions with CVD kernels and functions in the unit ball of L1.

Abstract

It is proved that interval quadrature formula of the form $$ q(f)=\sum\limits_{k=1}^nc_k\frac 1{2h}\int\limits_{x_k-h}^{x_k+h}f(t)dt $$ ($c_k\in \RR, \, x_1+h<x_2-h<x_2+h<...<x_n-h<x_n+h<x_1+2\pi -h$) with equal $c_k$ and equidistant $x_k$ is optimal among all such formulas for the class $K*F_1$ of convolutions of a $CVD$-kernel $K$ with functions from the unite ball of the space $L_1$ of $2\pi$-periodic integrable functions.