Generalized shift, generalized convolution and some extremal relations of the theory of approximation of function

We consider an operator of generalized shift defined on the space of functions summable on $[-1,1]$ with weight $(1-x)^\alpha(1+x)^\beta$ $(\alpha, \beta\geqslant-\frac12)$ and a generalized convolution associated with operator. Some differential operators are introduced and relevant classes of functions are considered, which can be represented in the form of generalized convolution, for these classes we obtain a number of extremal relations of the theory of approximation of function by algebraic polynomials. An essential role is played by some duality relations for classes of generalized convolutions.