On norms of Lewanowicz operators (Q1824791)

Let \(\Pi_ n\) be the subspace of all polynomials of degree less than or equal to n, and \({\mathcal P}_ n\) be the set of all linear projections P: C[-1,1]\(\to \Pi_ n\). If \((P_ n)_{n\in N}\) is a sequence of projections in \({\mathcal P}_ n\), the hitherto best known asymptotic equality is \(\| P_ n\| =\frac{4}{\pi^ 2}\ln n+O(1).\) Lewanowicz conjectured that it is possible to reduce the constant coefficient of \(\ell_ n n\). The author shows that this is impossible with Lewanowicz operators.