On generalized Lipschitz classes and Fourier series

In 1967 R.P. Boas Jr. found necessary and sufficient conditions of belonging of a function to a Lipschitz class. Later Boas's findings were generalized by many authors (M. and S. Izumi (1969), L.-Y. Chan (1991) and others). Recently, L. Leindler (2000) and J. Nemeth (2001) have published two papers, in which they have generalized all the previous results. The authors have considered the case, when the order of modulus of smoothness equals one (L. Leindler) or two (J. Nemeth). In this paper, we prove theorems of Boas-type for the modulus of smoothness of any order. Furthermore, we solve the inverse problem. Also, we discuss some conditions on a majorant which are equivalent to the well-known conditions of Bari-Stechkin.