1. UPPER BOUNDS OF APPROXIMATIONS BY FOURIER SUMS ON FUNCTION CLASSES

Nikol’skii’s early mathematical studies concerned the theory of linear operators in linear normed spaces. In this branch, he obtained profound results related to the validity of the Fredholm alter-native for linear equations. During the next years until 1951, Nikol’skii worked on various problems in the theory of ap-proximation of functions. This survey covers some of his fundamental results obtained during that period. Unfortunately, many of his other important results are beyond the scope of this paper. In this connection, we note Korneichuk’s large article devoted to the works by Nikol’skii in approxi-mation theory and their development [1]. In his 1951 work [2], Nikol’skii established inequalities in different metrics for trigonometric polynomials and entire functions of exponential type. These results laid a foundation for his own investigations and the studies by his numerous followers on the embedding theorems for spaces of differentiable functions of several variables and on their applications to problems of mathematical physics. In subsequent years, Nikol’skii’s investigations primarily concerned these subjects. They are surveyed in [3]