Про деякi новi критерiї нескiнченної диференцiйовностi перiодичних функцiй

Abstract

The set of D1 of infinitely differentiable periodic functions is studied in terms of generalized f derivatives defined by a pair f = ( f1, f2) of sequences f1 and f2. It is established that every function F from the set D1 has at least one such derivative whose parameters f1 and f2 decrease faster than any power function. For an arbitrary function from D1 different from a trigonometric polynomial, there exists a pair having the parameters f1 and f2 with the same properties, for which the f derivative already does not exist. On the basis of the proved statements, a number of criteria for a function to belong to the set D1 is given.