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.

Notation

Let be a Fourier series with Fourier coefficients , relating to each other as

such that the Toeplitz matrices are Hermitian, i.e., if then . Then both and eigenvalues are real-valued and the determinant of is given by

.

Szegő theorem

Under suitable assumptions the Szegő theorem states that

for any function that is continuous on the range of . In particular

 

 

 

 

(1)

such that the arithmetic mean of converges to the integral of .[4]

First Szegő theorem

The first Szegő theorem[1][3][5] states that, if right-hand side of (1) holds and , then

 

 

 

 

(2)

holds for and . The RHS of (2) is the geometric mean of (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Let be the Fourier coefficient of , written as

The second (or strong) Szegő theorem[1][6] states that, if , then

See also

References

  1. 1 2 3 Böttcher, Albrecht; Silbermann, Bernd (1990). "Toeplitz determinants". Analysis of Toeplitz operators. Berlin: Springer-Verlag. p. 525. ISBN 3-540-52147-X. MR 1071374.
  2. Ehrhardt, T.; Silbermann, B. (2001) [1994], "Szegö_limit_theorems", Encyclopedia of Mathematics, EMS Press
  3. 1 2 Simon, Barry (2011). Szegő's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials. Princeton: Princeton University Press. ISBN 978-0-691-14704-8.
  4. Gray, Robert M. (2006). "Toeplitz and Circulant Matrices: A Review" (PDF). Foundations and Trends in Signal Processing.
  5. Szegő, G. (1915). "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion". Math. Ann. 76 (4): 490–503. doi:10.1007/BF01458220. S2CID 123034653.
  6. 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.
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