Properties of Riemann sums for functions representable by a trigonometric series with monotone coefficients

We study properties of Riemann sums
$$ R_n(\varphi,a)=\frac{2\pi}n\sum_{k=0}^{n-1}\varphi\biggl(2\pi\frac{k+a}n\biggr),\qquad0\leqslant a\leqslant1, $$
for functions representable as the sum of a trigonometric series with monotone (or convex) coefficients. We consider two basic problems: 1) the connection between the behavior of these sums and the rate of decrease of the coefficients of the series; 2) the limit properties of the ratio of a coefficient of the series, considered as an integral, to a corresponding Riemann sum of higher order.
Bibliography: 4 titles.