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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Authors: Thomas Kriete and Tavan Trent
Journal: Proc. Amer. Math. Soc. 62 (1977), 83-88
MSC: Primary 46J15; Secondary 30A78
MathSciNet review: 0454643
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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 $|z| < 1$.

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Keywords: Measures on unit disk, <!– MATH ${H^2}(\mu )$ –> <IMG WIDTH="60" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${H^2}(\mu )$"> space, closure of polynomials, point evaluation functional, kernel function, functional Hilbert space, subnormal operator, growth estimates, Poisson integral
Article copyright: © Copyright 1977 American Mathematical Society