(untitled) (International Journal of Mathematics and Physics, 2016)
Assessment of the best approximations of functions through its modulus of smoothness (direct theorems of approximation theory or theorems of Jackson type) and of modulus of smoothness through its best approximation in a given metric (inverse theorems of approximation theory or theorems similar to Bernstein’s theorem) by orthogonal systems have been the subject of research of many generations of mathematicians. The relevance of this topic is determined by the numerical analysis, particularly with the development of computer technology. This topic is naturally linked to the further development of the tasks aiming to estimate the closest approximation of a function, approximation of inequalities of Bernstein and Nikolskiy, and of approximation theory [1-8].