Orthogonal polynomials of Sobolev type on the unit circle (Q1332293)

The authors consider a Hermitian form \(\langle,\rangle\) given by \[ \langle P(z), Q(z)\rangle= u(P(z) \overline{Q(z)})+ \lambda^{- 1} P'(a) \overline{Q'(a)}, \] where \(u\) is a regular linear functional which is Hermitian on the linear space of Laurent polynomials, \(\lambda\) is real, \(\lambda\neq 0\) and \(| a|= 1\). They find a necessary and sufficient condition for the existence of a sequence of monic orthogonal polynomials with respect to this form \(\langle,\rangle\). The special case of \(u\) being a positive definite functional is also considered. In that case asymptotic properties for the orthonormal polynomials with respect to \(\langle, \rangle\) are given.