Orthogonality properties of linear combinations of orthogonal polynomials II

Abstract

Let $(P_n)$ and $({\mathcal{P}}_n)$ be polynomials orthogonal on the unit circle with respect to the measures dσ and dµ, respectively. In this paper we consider the question how the orthogonality measures dσ and dµ are related to each other if the orthogonal polynomials are connected by a relation of the form $\sum\nolimits_{j = 0}^{k(n)} {\gamma _{j,n} {\mathcal{P}}_{n - j} (z)} = \sum\nolimits_{j = 0}^{l(n)} {\lambda _{j,n} P_{n - j} (z)}$, for $n \in {\mathbb{N}}$, where $\gamma _{j,n} ,\lambda _{j,n} \in {\mathbb{C}}$. It turns out that the two measures are related by $d\sigma \left( \phi \right) = {\mathcal{A}}\left( \phi \right)/{\mathcal{E}}\left( \phi \right)d\mu \left( \phi \right) + \sum M _j \delta \left( {e^{i\phi } - e^{i\kappa j} } \right)$ if $l\left( n \right) + k\left( n \right) \leqslant n/3$, where ${\mathcal{A}}$ and ${\mathcal{E}}$ are known trigonometric polynomials of fixed degree and where the $\kappa _j$'s are the zeros of ${\mathcal{E}}$ on $\left[ {0,\left. {2\pi } \right)} \right.$. If the $l\left( n \right)$'s and $k\left( n \right)$'s are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dµ have to be of the form ${\mathcal{A}}\left( \phi \right)/{\mathcal{S}}\left( \phi \right)d\phi$ and ${\mathcal{E}}\left( \phi \right)/{\mathcal{S}}\left( \phi \right)d\phi$, respectively, where ${\mathcal{A}},{\mathcal{E}},{\mathcal{S}}$ are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form $\Phi _n : = \sum\nolimits_{j = 0}^{l\left( n \right)} {\lambda _{j,n} P_{n - j} } + \sum\nolimits_{j = 0}^{l\left( n \right)} {\gamma _{j,n} } P_{_{n - j} }^* ,$ where $P_{_{n - j} }^* \left( z \right) = z^{n - j} \overline P _n \left( {1/z} \right)$ denotes the reciprocal polynomial of $P_{n - j}$, can be orthogonal.