A singular Riesz product in the Nevai class and inner functions with the Schur parameters in \(\cap_{p>2}I^p\) (Q5931921)

Consider a Riesz product \(\sigma\) on the unit circle in the complex plane. The corresponding Geronimus parameters \((a_n)_{n\geq 0}\) are defined by NEWLINE\[NEWLINEa_n= -\overline{\varphi_{n+1}(0)}/k_{n+ 1},NEWLINE\]NEWLINE where the \(\varphi_n= k_n z^n+\cdots+ \varphi_n(0)\) with \(k_n> 0\) are orthonormal polynomials with respect to \(d\sigma\).NEWLINENEWLINENEWLINEThe author proves the existence of a Riesz product \(\sigma\) with NEWLINE\[NEWLINE\sum^\infty_{n= 0}|a_n|^p< \infty\quad\text{for all }p> 2.NEWLINE\]NEWLINE According to a theorem of Geronimus, this also implies that the Schur parameters \((\gamma_n)_{n\geq 0}\) of the inner factor of the Cauchy integral of \(d\sigma\) satisfy the same condition, i.e., \((\gamma_n)\in\bigcap_{p> 2}\ell^p\).