ABSTRACT
ON CERTAIN EXPANSIONS IN BERNSTEIN POLYNOMIALS *
Introduction 1. The well-known Weierstrass theorem states that every continuous
function is the limit of a uniformly convergent sequence of polynomials. In 1912, Academician S. N. Bernstein (see [1]; [2], Add. to Ch. 5, pp. 120-126) gave a very interesting proof of this theorem based on the following idea:
If f ( x ) is a continuous function in the interval (0,1), then the sequence of polynomials
satisfies the assertion of the Weierstrass theorem. M.I. Chlodovsky [3] investigated the problem of convergence of the se
quence Pn(x) to a discontinuous function f (x) : in this case the convergence can be guaranteed only under very restrictive assumptions on the function
2. It is easily verified that every polynomial of degree n can be repre sented as
where <pk)n are appropriately chosen real numbers obtained by putting
and solving for
/(*)•
Polynomials written in the form (1) will be referred to as polynomials in the form of S. N. Bernstein. Bernstein’s theorem shows that this form of polynomials may prove very convenient.
