Szegő limit theorems

Szegő limit theorems In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices.[1][2][3] They were first proved by Gábor Szegő.

Contents 1 Notation 2 First Szegő theorem 3 Second Szegő theorem 4 References Notation Let φ : T→C be a complex function ("symbol") on the unit circle. Consider the n×n Toeplitz matrices Tn(φ), defined by {displaystyle T_{n}(phi )_{k,l}={widehat {phi }}(k-l),quad 0leq k,lleq n-1,} where {displaystyle {widehat {phi }}(k)={frac {1}{2pi }}int _{0}^{2pi }phi (e^{itheta })e^{-iktheta },dtheta } are the Fourier coefficients of φ.

First Szegő theorem The first Szegő theorem[1][4] states that, if φ > 0 and φ ∈ L1(T), then {displaystyle lim _{nto infty }{frac {det T_{n}(phi )}{det T_{n-1}(phi )}}=exp left{{frac {1}{2pi }}int _{0}^{2pi }log phi (e^{itheta }),dtheta right}.}         (1) The right-hand side of (1) is the geometric mean of φ (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem Denote the right-hand side of (1) by G. The second (or strong) Szegő theorem[1][5] asserts that if, in addition, the derivative of φ is Hölder continuous of order α > 0, then {displaystyle lim _{nto infty }{frac {det T_{n}(phi )}{G^{n}(phi )}}=exp left{sum _{k=1}^{infty }kleft|{widehat {(log phi )}}(k)right|^{2}right}.} References ^ Jump up to: a b c Böttcher, Albrecht; Silbermann, Bernd (1990). "Toeplitz determinants". Analysis of Toeplitz operators. Berlin: Springer-Verlag. p. 525. ISBN 3-540-52147-X. MR 1071374. ^ Ehrhardt, T.; Silbermann, B. (2001) [1994], "Szegö_limit_theorems", Encyclopedia of Mathematics, EMS Press ^ Simon, Barry (2010). Szegő's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials. Princeton: Princeton University Press. ISBN 978-0-691-14704-8. MR 1071374. ^ Szegő, G. (1915). "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion" (PDF). Math. Ann. 76 (4): 490–503. doi:10.1007/BF01458220. ^ Szegő, G. (1952). "On certain Hermitian forms associated with the Fourier series of a positive function". Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]: 228–238. MR 0051961. Categories: Theorems in analysis