On Banach systems (Q1100679)

A Banach system is an orthonormal system \(\{f_ n(t)\}\) on (0,1) for which all sums \(\sum^{nn}_{i=1}a_ if_ i(t)\) satisfy \(\int^{1}_{0}| P_ n(t)| dt\geq C\| P_ n(t)\|_ 2\) with C independent of \(P_ n\). Here it is shown that if \(\{\phi_ n(t)\}\) is an orthonormal system for which \(\int^{1}_{0}| \phi_ n(t)| dt\geq M>0\), then for every \(\epsilon >0\) there is a sequence of integers \(r_ k\) such that \(\{\phi_{r_ k}\}^{\infty}_{k=1}\) is a Banach system, where \(2^{ck^{1+\epsilon}}<r_ k\leq 2^{c(k+1)^{1+\epsilon}},\) so that the number of \(r_ k\leq N\) exceeds \(c'\log^{1/1+\epsilon}N\). It is stated that there is a similar result for the existence of a Sidon subsystem if \(0<m<| \phi_ n(t)| <M\).