We present a survey highlighting main aspects of the development of research dealing with solving 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 lp-norms of the sequences of Fourier coefficients are finite. In particular, our survey contains the available results on 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 \({\mathcal{S}}^{p}\) 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.
