A Generic Schur Function is an Inner One

Abstract

A Schur function sis a function which is holomorphic in an open unit disk $\mathbb{D}$ of the complex plane and is contractive there for |s(z)|≤1 for $z \in \mathbb{D}$. A Schur function is called exceptional if it is rational inner one. A contractive sequence w is a sequence w={γk}0≤<∞ of complex numbers satisfying the condition |γk| < 1 for every k. The Schur algorithm establishes a one-to-one correspondence between the set Ω of all contractive sequences w={γk}0≤<∞ and the set of all non-exceptional Schur functions. A sequence w ∈ Ώ is the sequence of the Schur parameters of the appropriate Schur function denoted as s _w . Using this Schur correspondence, we introduce a probability measure on the set Ώ, or, equivalently, on the set of all Schur functions. Namely, starting from an arbitrary probability measure μ on $\mathbb{D}$, we consider the set Ω as the set of sequences of independent identically distributed complex numbers from $\mathbb{D}$, with common distribution μ. (In other words, we introduce the product measure. We show that if the support of the measure μ consists of more than one point (otherwise there is no randomness), then p _μ almost every Schur function s _w is inner. If, in addition, the logarithmic integral converges: $\int\limits_\mathbb{D} {\ln \left( {1 - \left| \gamma \right|} \right)\mu \left( {d\gamma } \right) > - \infty }$, then for p _μ almost every Schur function the sequence of its Schur approximants converges pointwise almost everywhere (with respect to the Lebesgue measure) on the unit circle. The multiplicative ergodic theory is the main tool of investigation.