Résumé:

In 1967 R.P.Boas Jr. proved several classical results about necessary and sufficient conditions of belonging of functions to Lipschitz class. Later Boas's findings were generalized by many authors ( M. and S. Izumi (1969), L.-Y. Chan (1991), L. Leindler (2000), J. Nemeth (2001)). In these works the cases were considered, when the order of modulus of smoothness equals one or two. In this talk, we present the theorems of Boas-type for modulus of smoothness of any order. Also, we consider the inverse problem. Furthermore, we provide some conditions on a majorant which are equivalent to well-known conditions of Bari-Stechkin.