Actual problems of the theory of approximations in metrics of discrete spaces on the sets of summable periodic and almost periodic functions

• Datum: 2025
• Hauptverfasser: Serdyuk, A., Shidlich, A., Сердюк, Анатолій, Шидліч, Андрій, Shidlich, Andrii
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2025
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/8525
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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 ${\mathcal S}^p$ and $B{\mathcal S}^p$ 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 $\psi$-derivatives and generalized moduli of smoothness in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p.$ Special attention is given to the development of investigations related to the derivation of direct and inverse approximation theorems in these spaces.