  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.
