Growth near the boundary in $H^{2}(\mu )$ spaces

Abstract:

Let ${H^2}(\mu )$ be the closure in ${L^2}(\mu )$ of the complex polynomials, where $\mu$ is a finite Borel measure supported on the closed unit disk in the complex plane. For $|z| < 1$, let $E(z) \equiv \sup |p(z)|/\left \| p \right \|$ where the supremum is over all polynomials p whose ${L^2}(\mu )$ norm $\left \| p \right \|$ is nonzero. An inequality is derived asymptotically relating $E(z)$ (as z tends to the unit circle) to the part of $\mu$ supported on the unit circle. The interplay between $\mu$ and the growth of functions in ${H^2}(\mu )$ is studied in the event that $E(z) < \infty$ for $|z| < 1$.