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

Анотація

Let $f\in C^{\infty}[-1,1]$ and $\exists \rho\in [1,2)$ such that $\forall k=0,1,2,\ldots$ $||f^{(k)}||_{C_{[-1,1]}}\le$ $ 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 $|f^{(n)}(x_{n,k})|\le C A(n),$  $\frac{A(n+1)}{A(n)}\le 2^{n+1/2}$ 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.