Szegő limit theorems - Alchetron, The Free Social Encyclopedia

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

Notation

Let φ : T→C be a complex function ("symbol") on the unit circle. Consider the n×n Toeplitz matrices T_n(φ), defined by

T n ( ϕ ) k , l = ϕ ^ ( k − l ) , 0 ≤ k , l ≤ n − 1 ,

where

ϕ ^ ( k ) = 1 2 π ∫ 0 2 π ϕ ( e i θ ) e − i k θ d θ

are the Fourier coefficients of φ.

First Szegő theorem

The first Szegő theorem states that, if φ > 0 and φ ∈ L₁(T), then

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 asserts that if, in addition, the derivative of φ is Hölder continuous of order α > 0, then

lim n → ∞ det T n ( ϕ ) G n ( ϕ ) = exp ⁡ { ∑ k = 1 ∞ k | ( log ⁡ ϕ ) ^ ( k ) | 2 } .

References

Szegő limit theorems Wikipedia

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Contents

Notation

First Szegő theorem

Second Szegő theorem

References