S M Nikolskii's plenary: ICM 1954

In September 1954 the International Congress of Mathematicians was held in Amsterdam. Sergei Mikhailovich Nikolskii was a plenary speaker delivering the lecture Einige Fragen der Approximation von Funktionen durch Polynome. We present below an English translation of Nikolskii's Introduction to his lecture.

Some questions concerning the approximation of functions by polynomials

Introduction

There are two periods in the history of theoretical studies devoted to the approximation of functions by polynomials.

In the previous century, during the first period, the efforts of mathematicians were directed towards elaborating approximation methods for individual functions. However, the foundations for the current approximation theory of functions were also laid at the time.

P L Chebyshev introduced the concept of the best approximation and discovered the theorem with the help of which we recognise whether a given continuous function on a segment is approximated by the polynomial in the best way or not.

Chebyshev, as well as his pupils and successors, also examined the special properties of the polynomials, the so-called extremal properties, which play a role as a proof tool in the modern approximation theory of functions. The well-known Markov inequalities for algebraic polynomials and the Bernstein inequality for trigonometric polynomials are examples of these properties.

The second period is already in the present time. It was formed in the beginning of our century after Weierstrass proved his theorem, which established the basis possibility of approximation on a segment of any continuous function using polynomials.

While the approximation of individual functions was the object of investigation in the previous century, the current focus is on questions of approximation of classes of functions.

The class of functions is given that is analytical, differentiable (in some way normalisable), sufficient for the Lipschitz condition, etc. The approximation method of the functions of this class is also given by polynomials. It is required that everything possible is said about the size or the change character of the approximations of the functions of this class.

The aim of this lecture is to provide a brief overview of some of the results related to this topic that have been obtained in the past twenty years. Rather, attention is focused only on questions with which I myself had close contact in my personal investigations.

Last Updated April 2020