On the second derivative of the sums of trigonometric series

Abstract

Some representations for the second derivatives of the sums of the cosine or sine trigonometric series are found in terms of the second differences of their coefficients. If for the cosine series we denote its sum by $f(x)$, then it is proved that under certain conditions the function $f'(x)-(a_{1}-2a_{2})x$ is concave or convex on $(0,\pi]$, which demonstrates an adherence of those representations. Also, we have obtained some estimates of the integrals of the absolute values of those derivatives in terms of the coefficients of such series.