On One Problem of Optimization of Approximate Integration

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.