Orthogonality and Boundary Interpolation

Abstract

Let $\left\{ {{\alpha _n}:n = 1,2,\ldots } \right\}$ be a sequence of (not necessarily distinct) points on the unit circle. Set ${\omega _0}(z) \equiv 1,{\omega _n}(z) = (z - {\alpha _1})(z - {\alpha _2}) \ldots (z - {\alpha_n})for n = 1,2,\ldots,$and let G denote the linear space spanned by the functions$\left\{ {1/{\omega _n}:n = 0,1,2,\ldots,} \right\}$.LetMbe a quasi-definite linear functional on$\mathcal{L} \cdot \mathcal{L}$, and define the inner product$, by f,g = M(f{g_ * })$, where $g * (z) = \overline {g(1\left){\vphantom{1z}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{z}})}$. (In particular M may be a positive definite functional given by$M(f) = \int_{ - \pi }^\pi {f({e^{it}})d\mu (t)},$,where µ is a measure such that all functions in $\mathcal{L} \cdot \mathcal{L}$ are µ-integrable).Let $\left\{ {{\varphi _n}} \right\}$be an orthogonal system obtained from the basis $\left\{ {1/{\omega _n}} \right\}$ by the Gram-Sdunidt method and define the associated

functions$\left\{ {{\varphi _n}} \right\}$ by ${\varphi _n}(z) = M(\frac{{t + z}}{{t - z}}\left[ {{\varphi _n}(t) - {\varphi _n}(z)} \right])$.Interpolation of ${\psi _n}/{\varphi _n}$to $M(\frac{{t + z}}{{t - z}})$ (or to the formal series

${\mu _0} + 2\sum\nolimits_{m = 1}^\infty {{\mu _m}z{\omega _{m - 1}}} (z),{\mu _m} = M(1/{\omega _m})$ at interpolation points ${\alpha _k}$is

discussed.