Difference between revisions of "Beurling theorem"
From Encyclopedia of Mathematics
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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 | + | 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=18587
Beurling theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beurling_theorem&oldid=18587
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article