Actual problems of the approximation theory in metrics of discrete spaces on sets of summable periodic and almost periodic functions
<|reference_start|>Actual problems of the approximation theory in metrics of discrete spaces on sets of summable periodic and almost periodic functions: This review paper highlights 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 review contains the results known so far concerning the best, best $n$-term approximations and widths of classes of functions of one and many variables defined by means of $\psi$-derivatives and generalized moduli of smoothness in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p$. Particular attention is paid to the development of studies related to the derivation of direct and inverse approximation theorems in these spaces.<|reference_end|>
@article{serdyuk2024actual,
    title={Actual problems of the approximation theory in metrics of discrete
  spaces on sets of summable periodic and almost periodic functions},
    author={Anatolii Serdyuk, Andrii Shidlich},
    journal={arXiv preprint arXiv:2407.04329},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.04329},
    primaryClass={math.CA cs.NA math.NA}
}