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Difference between revisions of "Beurling theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301301.png" /> be a function in the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301302.png" /> (cf. also [[Hardy classes|Hardy classes]]). The vector space spanned by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301304.png" />, is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301305.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301306.png" /> is an outer function (cf. also [[Hardy classes|Hardy classes]]).
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Let $f$ be a function in the Hardy class $H^2$ (cf. also [[Hardy classes|Hardy classes]]). The vector space spanned by the functions $e^{in\theta}f$, $n\geq0$, is dense in $H^2$ if and only if $f$ is an outer function (cf. also [[Hardy classes|Hardy classes]]).
  
This follows from the characterization of closed shift-invariant subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301307.png" /> as being of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301308.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130130/b1301309.png" /> an inner function.
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This follows from the characterization of closed shift-invariant subspaces in $H^2$ as being of the form $gH^2$ with $g$ an inner function.
  
 
See [[Beurling–Lax theorem|Beurling–Lax theorem]] for further developments.
 
See [[Beurling–Lax theorem|Beurling–Lax theorem]] for further developments.

Latest revision as of 14:34, 7 October 2014

Let $f$ be a function in the Hardy class $H^2$ (cf. also Hardy classes). The vector space spanned by the functions $e^{in\theta}f$, $n\geq0$, is dense in $H^2$ if and only if $f$ is an outer function (cf. also Hardy classes).

This follows from the characterization of closed shift-invariant subspaces in $H^2$ as being of the form $gH^2$ with $g$ an inner function.

See Beurling–Lax theorem for further developments.

References

[a1] B. Sz.-Nagy, C. Foias, "Harmonic analysis of operators on Hilbert spaces" , North-Holland (1970) pp. 104
[a2] B. Beauzamy, "Introduction to operator theory and invariant subspaces" , North-Holland (1988) pp. 194
[a3] W. Mlak, "Hilbert spaces and operator theory" , Kluwer Acad. Publ. (1991) pp. 188; 190
How to Cite This Entry:
Beurling theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beurling_theorem&oldid=33509
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article