This is a project in harmonic analysis and approximation theory.We present research actions in the following three topics, which are closely related to each other:1. Generalised Lipschitz class and Fourier series. 2. Moduli of smoothness of fractional orde r. 3.Relationship between Besov, Nikolsky, Besov-Nikolsky and Weyl-Nikolsky classes of functions. In the first topic, the objective consists in finding necessary and sufficient conditions on the Fourier coefficients of a function to belong to a generalised Lipschitz class. These types of results are, nowadays, known as Boas-type results, since it was R.P. Boas who in 1967 proved the first characterisation of this type. Since then, this theory has been widely studied by several authors (M. and S. Izumi, 1969 ; J. Nemeth, 1990; L Leindler, 2000). Their papers contain Boas-type results for a particular class of Lipschitz function and use the moduli of smoothness only of order one and two. The goal of the project consists in extending this result to a very genera l Lipschitz class and proving the corresponding results for moduli of smoothness of any order. The main object to study in the second topic is the concept of modulus of smoothness of fractional order. The first objective is to give the description of smoot hness modulus of fractional order and to find its decrease to zero's order for a certain class of functions. The second goal of the project in this topic is to prove direct theorems of approximation theory for constrained approximation. Here the concept of averaged moduli of smoothness of fractional order and fractional K-functionals will be used. Finally, the third topic, which is connected with the estimate of the moduli of smoothness previously presented, consists in studying the relationship between dif ferent classes of functions such as Besov, Nikolsky, Besov-Nikolsky and Weyl-Nikolsky classes, together with the problem of finding the corresponding interpolation spaces.
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