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factorization theorem for $H^\infty$ functions (Theorem)

Let $H^\infty$ denote the bounded analytic functions on the unit disc.

Theorem 1   Every $f \in H^\infty$ can be written as \begin{equation*} f(z) = \alpha I(z) F(z) \end{equation*}where $\lvert \alpha \rvert = 1$ $I$ is an inner function and $F$ is a bounded outer function. Conversely any function which can be so written is bounded.

Bibliography

1
John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.




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See Also: inner function

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Cross-references: function, conversely, outer function, inner function, unit disc, analytic functions, bounded
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This is version 1 of factorization theorem for $H^\infty$ functions, born on 2005-12-07.
Object id is 7523, canonical name is FactorizationTheoremForHinftyFunctions.
Accessed 1853 times total.

Classification:
AMS MSC30H05 (Functions of a complex variable :: Spaces and algebras of analytic functions)

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