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

One considers the generalized shift operator, defined on the space of functions, summable on –1, 1] with the weight (1–x)(1+x)(–1/2), and the corresponding generalized convolution operator. One introduces some differential operators and one considers the corresponding classes of functions, which can be represented in the form of a generalized convolution. For these classes one obtains a series of extremal relations of the theory of approximation of functions by algebraic polynomials. An essential role is played by some duality relations for the classes of generalized convolutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 150–157, 1986.