Abstract
This chapter discusses theorems connecting the best approximations of functions with their structural properties. It discusses some of the examples with specialized character and the exact order of decrease of the best approximations of functions with the type of singularity. The basic idea of the proof of the classical theorem of Weierstrass is based on replacing the graph of the continuous function approximated on [a, b] by a polygonal line inscribed in it. The best uniform approximation of functions has a discontinuous derivative of bounded variation. It plays a fundamental part in the study of the asymptotic properties of the best approximation by polynomials of other functions with singularities of the same type.
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