Actual problems of the theory of approximations in metrics of discrete spaces on the sets of summable periodic and almost periodic functions - Serdyuk, A.; Shidlich, A. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.5 We present a review highlighting the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces S^p and BS^p of periodic and almost periodic summable functions, respectively, where the l_p-norms of the sequences of Fourier coefficients are finite. In particular, the present review contains the available results concerning the best and best n-term approximations, as well as the widths of the classes of functions of one and many variables defined by means of the ψ-derivatives and generalized moduli of smoothness in the spaces S^p and BS^p. Special attention is given to the development of investigations related to the derivation of direct and inverse approximation theorems in these spaces.