Inequalities connected with weighted minimax series (Q1572810)

Let \(f\) be a continuous function having zero point \(x_0\). Let denote \(E_n= E_n(f)\) the best approximation of \(f\) by polynomials of degree \(\leq n\), and \[ N_r(f):= E_0+ \sum^\infty_{k=1} k^r E_k,\quad r>0. \] The main result states that \(N_r(fg)\leq \alpha_r N(f)N(g)\), where \(\alpha_r\) is independent of \(x_0\). Using the new results certain Banach algebras are also defined.