Approximation of convolution classes (Q908463)

Until recently the problems of the best and the best one-sided approximations were studied separately. In this paper a number of new results solving these problems are obtained, and it is proved that practically all known results can be included into the frame of one theory, which is essentially complete and general owing to the vast variety of the considered function classes. In the first part of this article we present the results on the best (\(\alpha\),\(\beta)\)- approximation in \(L_ 1\) of kernels by trigonometric polynomials, we extend the Nikol'skij theory of the best approximation in mean of the convolution classes onto nonsymmetric (\(\alpha\),\(\beta)\)-approximations, and we show that the resulting theory can be applied to a very wide collection of the kernels K and the function classes \(K*F_{1;\gamma,\delta}\), \(\gamma,\delta >0\). The best (\(\alpha\),\(\beta)\)-approximations of the kernels K are used in Section 3 to study the problem of the best linear approximation of the classes \(K*F_{\infty;\gamma,\delta}\) in the space C. The results of the second part of this article concern the classes of convolutions with kernels, which do not increase the number of sign changes (CVD- or CVD[\(\Delta\) ]-kernels; for the strict definition see Section 5). The restriction, in comparison with the first part of the article, of the set of considered kernels K permits us to enlarge the collection of the spaces F, from which the functions convoluted with K are taken (as F are taken arbitrary transpose invariant sets; for the definitions and examples see Section 8). Therefore, besides the problems of approximations by trigonometric polynomials, it is also possible to study the problems of approximations by the convolutions of the kernel K with \(2\pi\)-periodic polynomial splines. In the derivation of the results of the second part of this article an essential role plays the rather wide generalization of the known inequalities of the Bohr-Favard-Hörmander type for functions orthogonal to the trigonometric polynomials, and their analogs for the functions orthogonal to splines.