View source for Beurling theorem

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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 $H^2$ as being of the form $gH^2$ with $g$ an inner function.

See [[Beurling–Lax theorem|Beurling–Lax theorem]] for further developments.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Sz.-Nagy,   C. Foias,   "Harmonic analysis of operators on Hilbert spaces" , North-Holland  (1970)  pp. 104</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Beauzamy,   "Introduction to operator theory and invariant subspaces" , North-Holland  (1988)  pp. 194</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Mlak,   "Hilbert spaces and operator theory" , Kluwer Acad. Publ.  (1991)  pp. 188; 190</TD></TR></table>