On One Problem of Optimization of Approximate Integration

Abstract

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