Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

Abstract

Listen

Translate article

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersMk such that, for anyf(z)ec[0,1] and ∀ > 0 there exists the polynomial\(P\left( z \right) = \sum\nolimits_0^n {\lambda _k z^k } \) that ∥f−P∥<e and ∣λk∣<eMk,k=1, ...,n.