Best trigonometric approximation and modulus of smoothness of functions in weighted grand Lebesgue spaces

In this work, first of all, L-omega(p),theta) (T) weighted grand Lebesgue spaces and Muckenhoupt weights is defined. The information about properties of these spaces is given. Let T-n be the trigonometric polynomial of best approximation. The approximation of the functions in grand Lebesgue spaces have been investigated by many authors. In this work the relation between fractional derivatives of a T-n trigonimetric polynomial and the best approximation of the function is investigated in weighted grand Lebesgue spaces. In that regard, the neccessary and sufficient condition is expressed in Theorem 1. In addition, in this work in weighted grand Lebesgue spaces a specific operator is defined. Later on, with the help of this operator the fractional modules of smoothness of order r of function f is defined. Also, in this work, using the properties of modulus of smoothness of function, the relationship between the fractional modulus of smoothness of the function and n - th partial and de la Vallee-Poussin sums of its Fourier series in subspace of weighted grand Lebesgue spaces are studied. These results are expressed in Theorem 2.