Differential properties of sums of trigonometric series of a certain class (Q1584108)

The authors consider the series \[ \begin{gathered} \frac{a_0}{2} + \sum_{k=1}^\infty a_k \cos kx,\tag{1}\\ \sum_{k=1}^\infty a_k \sin kx,\tag{2} \end{gathered} \] whose coefficients \(u_k\) tend to zero. The series (1) and (2) converge on each segment \( [\varepsilon,\pi]\), \( \varepsilon>0\). It is assumed that for some \( \sigma\geq 0 \) and natural \(r\) the series \[ \sum_{k=1}^\infty k^\rho|\Delta^\sigma a_k| \] is convergent, where \(\rho\) and \(\sigma\) are given nonnegative numbers and \[ \Delta^\sigma a_k := \sum_{m=0}^\infty \begin{pmatrix} m-\sigma-1\\ m\end{pmatrix} a_{k+m}. \] It is proved that the sums of these series have continuous derivatives of the order \([\rho]\) on \( (0,\pi]\), but may not have the derivatives of the order \( |\rho|+1 \) in any point.