A Losynski-Kharshiladze theorem for Müntz polynomials (Q1073278)

The second author of the present note has conjectured that for any sequence of linear bounded projections \(p_ n: C[0,1]\to C[0,1]\) with range \(p_ n=span\{t^{\lambda_ j}\}^ n_{j=0}\) we have \(\| p_ n\| \to \infty\), where \(\{\lambda_ j\}_ 0^{\infty}\) is an increasing sequence of integers such that \(\lambda_ 0=0\). This conjecture turned out to be false in general. In this note the authors indicate some sequences for which the conjecture is true and some for which it is false. Thus they prove that if \(\{\lambda_ i\}\) is a lacunary sequence, then there exists a sequence of projections \(p_ n: C[0,1]\to C[0,1]\) with range \(p_ n=span\{t^{\lambda_ i}\}^ n_{i=0}\) such that \(\| p_ n\| =O(1)\). On the other hand if \(\lambda_ n\leq n+o(\log n)\), then for every sequence of projections \(p_ n: C[0,1]\to C[0,1]\) with range \(p_ n=span\{t^{\lambda_ i}\}^ n_{i=0}\), \(\| p_ n\| \to \infty\).