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Proceedings of the American Mathematical Society

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



Growth near the boundary in $ H\sp{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 $ \vert z\vert < 1$, let $ E(z) \equiv \sup \vert p(z)\vert/\left\Vert p \right\Vert$ where the supremum is over all polynomials p whose $ {L^2}(\mu )$ norm $ \left\Vert p \right\Vert$ 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 $ \vert z\vert < 1$.

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Keywords: Measures on unit disk, $ {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