On a relation between the coefficients and the sum of the generalized Taylor series

Let $f\in C^\infty [-1,1]$ and $\exists\,\rho\in [1,2)$ such that $\forall\,k=0,1,2,\dots$ $\|f^{(k)}\|_{C[-1,1]}\leq c(f)\rho^k2^{\frac{k(k+1)}2}$. Then it expands in the generalized Taylor series, which was introduced by V. A. Rvachov in 1982. In this paper it is shown that if the restrictions $\|f^{(n)}\|=o(2^{\frac{n(n+1)}2})$, $n\to\infty$ are imposed on the sum of this series, and stronger restrictions $|f^{(n)}(x_{n,k})|\leq CA(n)$, $\frac{A(n+1)}{A(n)}\leq 2^{n+\frac 12} $ hold for its coefficients, then these stronger restrictions will hold for the sum of the series too. As a consequence the conditions of belonging to Gevrey class and of real analyticity for the above-mentioned functions are obtained.