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

Анотація

Let fÎ C¥[-1, 1] and $ r &Icirc [1, 2) such that "k = 0, 1, 2, .... ||f ^(k)||_C[-1,1] ≤ c(f)r^k2. 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), n → 1 are imposed on the sum of this series, and stronger restrictions |f ⁽ⁿ⁾(x_n,k)| ≤ CA(n), ≤ 2^n+½ 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.