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The sharp form of the strong Szego theorem | | Barry Simon
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6 Feb 2004 | | Subject: | Spectral Theory | math.SP | | Abstract: | Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1) imes (n+1)$ matrix with elements ${c_{j-i}}_{0leq i,jleq n}$ where $c_m =hat f_mequiv int e^{-im heta} f( heta) f{d heta}{2pi}$. The sharp form of the strong SzegH{o} theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (f{d heta}{2pi})$, we have lim_{n oinfty} D_n(e^L) e^{-(n+1)hat L_0} = exp iggl(sum_{k=1}^infty kabs{hat L_k}^2iggr) where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{i heta} o L( heta)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained. | | Source: | arXiv, math.SP/0402110 | | Services: | Forum | Review | PDF | Favorites | |
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