On universal functions and series (Q1093823)

In the first part of the paper the non-density of the difference quotients is proved in \(L^ p\) spaces when \(p\geq 1\). In the second part universal series are studied. Suppose that \(\phi_ i\) is a complete orthonormal system in \(L^ 2\). The series \(\sum^{+\infty}_{1}c_ i\phi_ i\) is called universal in \(L^ p\) if its partial sums are dense in \(L^ p\). The effect of Toeplitz summation on the universality is investigated. Roughly speaking: if \(p\geq 1\) then there are no universal series in the Toeplitz sense while for \(0\leq p<1\) ``typical'' series are universal. Finally, as a by-product, it is proved that if \(\phi_ n\) is a complete orthonormal system in \(L^ 2[0,1]\) then \(\sum^{+\infty}_{n=1}\| \phi_ n\|^ 2_ 1=+\infty\).