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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306401.png" /> be a complex-valued function defined on the complex unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306402.png" />, with [[Fourier coefficients|Fourier coefficients]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306403.png" /></td> </tr></table>
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Out of 75 formulas, 73 were replaced by TEX code.-->
  
Szegö limit theorems describe the behaviour of the determinants of the Toeplitz matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306404.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306405.png" /> tends to infinity, for certain classes of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306406.png" /> (cf. also [[Toeplitz matrix|Toeplitz matrix]]).
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Let $a$ be a complex-valued function defined on the complex unit circle $\bf T$, with [[Fourier coefficients|Fourier coefficients]]
  
For real positive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306407.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306408.png" />, G. Szegö [[#References|[a7]]] has proved that
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\begin{equation*} a _ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } a ( e ^ { i \theta } ) e ^ { - i n \theta } d \theta. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s1306409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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Szegö limit theorems describe the behaviour of the determinants of the Toeplitz matrices $T _ { n } ( a ) = ( a _ { j - k } ) _ { j , k = 0 } ^ { n - 1 }$, as $n$ tends to infinity, for certain classes of functions $a$ (cf. also [[Toeplitz matrix|Toeplitz matrix]]).
  
with the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064010.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064011.png" /> stands for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064012.png" />th Fourier coefficient of the logarithm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064013.png" />. A statement of type (a1) is referred to as a first Szegö limit theorem. Szegö's result has been considerably extended. In particular, (a1) holds for functions that are the exponentials of continuous complex-valued functions defined on the unit circle.
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For real positive functions $a \in L ^ { 1 } ( {\bf T} )$ for which $\operatorname { log } a \in L ^ { 1 } (\bf T )$, G. Szegö [[#References|[a7]]] has proved that
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\begin{equation} \tag{a1} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { \operatorname { det } T _ { n - 1 } ( a ) } = G ( a ), \end{equation}
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with the constant $G ( a ) = \operatorname { exp } ( [ \operatorname { log } a ] _ { 0 } )$. Here, $[ \operatorname { log } a ] _ { k }$ stands for the $k$th Fourier coefficient of the logarithm of $a$. A statement of type (a1) is referred to as a first Szegö limit theorem. Szegö's result has been considerably extended. In particular, (a1) holds for functions that are the exponentials of continuous complex-valued functions defined on the unit circle.
  
 
The strong Szegö limit theorem states that
 
The strong Szegö limit theorem states that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { G ( a ) ^ { N } } = E ( a ), \end{equation}
  
with the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064015.png" /> defined by
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with the constant $E ( a )$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064016.png" /></td> </tr></table>
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\begin{equation*} E ( a ) = \operatorname { exp } \left( \sum _ { k = 1 } ^ { \infty } k [ \operatorname { log } a ] _ { k } [ \operatorname { log } a ]_{ - k} \right). \end{equation*}
  
 
Relation (a2) was first proved by Szegö [[#References|[a8]]] for positive real functions whose derivatives satisfy a Hölder–Lipschitz condition. This result has been generalized too. For instance, the strong Szegö limit theorem holds for functions that are the exponentials of continuous and sufficiently smooth complex-valued functions defined on the unit circle.
 
Relation (a2) was first proved by Szegö [[#References|[a8]]] for positive real functions whose derivatives satisfy a Hölder–Lipschitz condition. This result has been generalized too. For instance, the strong Szegö limit theorem holds for functions that are the exponentials of continuous and sufficiently smooth complex-valued functions defined on the unit circle.
  
Such results about the asymptotics of Toeplitz determinants can be used to obtain information about the asymptotic distribution of the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064017.png" /> of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064018.png" />. It turns out that
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Such results about the asymptotics of Toeplitz determinants can be used to obtain information about the asymptotic distribution of the eigenvalues $\{ \lambda _ { k } ^ { ( n ) } \} _ { k = 1 } ^ { n }$ of the matrices $T _ { n } ( a )$. It turns out that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f ( \lambda _ { k } ^ { ( n ) } ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } f ( a ( e ^ { i \theta } ) ) d \theta + o ( 1 ), \end{equation}
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064020.png" />, if, for instance, one of the following assumptions is satisfied:
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as $n \rightarrow \infty$, if, for instance, one of the following assumptions is satisfied:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064021.png" /> is real-valued and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064022.png" /> is a [[Continuous function|continuous function]] on the real line with a compact support [[#References|[a9]]];
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$a \in L ^ { 1 } ( {\bf T} )$ is real-valued and $f$ is a [[Continuous function|continuous function]] on the real line with a compact support [[#References|[a9]]];
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064023.png" /> is a continuous complex-valued function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064024.png" /> is an [[Analytic function|analytic function]] defined on an open neighbourhood of the set
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$a$ is a continuous complex-valued function and $f$ is an [[Analytic function|analytic function]] defined on an open neighbourhood of the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064025.png" /></td> </tr></table>
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\begin{equation*} \operatorname{spec} T ( a ) = \operatorname { Ran } ( a ) \bigcup \{ z \notin \operatorname { Ran } ( a ) : \text { wind } ( a - z ) \neq 0 \}. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064026.png" /> stands for the [[Toeplitz operator|Toeplitz operator]] acting on the [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064028.png" /> refers to its spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064029.png" /> stands for the range of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064030.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064031.png" /> denotes the [[Winding number|winding number]] of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064032.png" />. The asymptotic formula (a3) is sometimes also called the first Szegö limit theorem or a first-order trace formula. A second-order trace formula, which is the pendant of the strong Szegö limit theorem, has also been established [[#References|[a2]]], [[#References|[a10]]].
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Here, $T ( a ) = ( a _ { j  - k} ) _ { j , k = 0} ^ { \infty }$ stands for the [[Toeplitz operator|Toeplitz operator]] acting on the [[Hilbert space|Hilbert space]] $\text{l} ^ { 2 }$, $\operatorname {spec} T ( a )$ refers to its spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]), $\operatorname{Ran}( a )$ stands for the range of the function $a$, and $\operatorname{wind}( a - z )$ denotes the [[Winding number|winding number]] of the function $a ( e ^ { i \theta } ) - z$. The asymptotic formula (a3) is sometimes also called the first Szegö limit theorem or a first-order trace formula. A second-order trace formula, which is the pendant of the strong Szegö limit theorem, has also been established [[#References|[a2]]], [[#References|[a10]]].
  
Some work was also done in order to determine the higher-order terms of the [[Asymptotic expansion|asymptotic expansion]] of Toeplitz determinants [[#References|[a1]]]. Exact formulas for Toeplitz determinants in terms of the Wiener–Hopf factorization (cf. also [[Wiener–Hopf method|Wiener–Hopf method]]; [[Wiener–Hopf operator|Wiener–Hopf operator]]) of the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064033.png" /> do also exist (see, e.g., [[#References|[a11]]]).
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Some work was also done in order to determine the higher-order terms of the [[Asymptotic expansion|asymptotic expansion]] of Toeplitz determinants [[#References|[a1]]]. Exact formulas for Toeplitz determinants in terms of the Wiener–Hopf factorization (cf. also [[Wiener–Hopf method|Wiener–Hopf method]]; [[Wiener–Hopf operator|Wiener–Hopf operator]]) of the generating function $a$ do also exist (see, e.g., [[#References|[a11]]]).
  
H. Widom [[#References|[a10]]] was the first to give a crystal clear proof of the strong Szegö limit theorem, by an elegant application of ideas from operator theory and thereby replacing earlier long-winded proofs. With his approach he was able to generalize this theorem to the case of matrix-valued functions. Under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064034.png" /> is a sufficiently smooth matrix-valued function defined on the unit circle for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064035.png" /> is the exponential of a continuous function, (a2) still holds, but with constants defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064037.png" />. The last expression has to be understood as an operator determinant. In this connection, the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064038.png" /> plays an important role, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064039.png" /> is a [[Hankel operator|Hankel operator]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064040.png" />. Note that for sufficiently smooth and invertible matrix functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064041.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064042.png" /> is a trace-class operator (cf. also [[Nuclear operator|Nuclear operator]]). An explicit expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064043.png" /> is not known yet (as of 2000), apart from special cases related to the scalar situation. On the other hand, an operator-valued version of the strong Szegö limit theorem has been established [[#References|[a3]]].
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H. Widom [[#References|[a10]]] was the first to give a crystal clear proof of the strong Szegö limit theorem, by an elegant application of ideas from operator theory and thereby replacing earlier long-winded proofs. With his approach he was able to generalize this theorem to the case of matrix-valued functions. Under the assumption that $a$ is a sufficiently smooth matrix-valued function defined on the unit circle for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064035.png"/> is the exponential of a continuous function, (a2) still holds, but with constants defined by $G ( a ) = \operatorname { exp } ( [ \operatorname { log } \operatorname { det } a ] _ { 0 } )$ and $E ( a ) = \operatorname { det } T ( a ) T ( a ^ { - 1 } )$. The last expression has to be understood as an operator determinant. In this connection, the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064038.png"/> plays an important role, where $H ( a ) = ( a _ { 1  + j + k} )_{ j,k = 0}^{\infty}$ is a [[Hankel operator|Hankel operator]] and $\tilde { a } ( e ^ { i \theta } ) = a ( e ^ { - i \theta } )$. Note that for sufficiently smooth and invertible matrix functions $a$ the operator $H ( a ) H ( \tilde{a} ^ { - 1 } )$ is a trace-class operator (cf. also [[Nuclear operator|Nuclear operator]]). An explicit expression for $E ( a )$ is not known yet (as of 2000), apart from special cases related to the scalar situation. On the other hand, an operator-valued version of the strong Szegö limit theorem has been established [[#References|[a3]]].
  
The asymptotic behaviour of Toeplitz determinants changes considerably if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064044.png" /> is discontinuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064045.png" /> possesses zeros, poles, jumps, or certain oscillations, then the asymptotics is predicted by the Fisher–Hartwig conjecture or by the more general Basor–Tracy conjecture. Let
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The asymptotic behaviour of Toeplitz determinants changes considerably if the function $a$ is discontinuous. If $a$ possesses zeros, poles, jumps, or certain oscillations, then the asymptotics is predicted by the Fisher–Hartwig conjecture or by the more general Basor–Tracy conjecture. Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064046.png" /></td> </tr></table>
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\begin{equation*} a ( e ^ { i \theta } ) = b ( e ^ { i \theta } ) \prod _ { r = 1 } ^ { R } \omega _ { \alpha _ { r } , \beta _ { r } } ( e ^ { i ( \theta - \theta _ { r } ) } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064047.png" /> are distinct points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064048.png" /> is the exponential of a sufficiently smooth function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064050.png" /> are complex parameters. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064051.png" /> is defined as
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where $\theta _ { 1 } , \dots , \theta _ { R } \in [ 0,2 \pi )$ are distinct points, $b$ is the exponential of a sufficiently smooth function and $\alpha_r$, $\beta _ { r }$ are complex parameters. The function $\omega _ { \alpha , \beta }$ is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064052.png" /></td> </tr></table>
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\begin{equation*} \omega _ { \alpha , \beta } ( e ^ { i \theta } ) = ( 2 - 2 \operatorname { cos } \theta ) ^ { \alpha } e ^ { i \beta ( \theta - \pi ) } , 0 &lt; \theta &lt; 2 \pi . \end{equation*}
  
 
Then the Fisher–Hartwig conjecture [[#References|[a6]]] asserts that
 
Then the Fisher–Hartwig conjecture [[#References|[a6]]] asserts that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064053.png" /></td> </tr></table>
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\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { G ( b ) ^ { n } n ^ { \Omega } } = E, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064054.png" />. An explicit, but more complicated expression is known for the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064055.png" />. It has turned out that in some cases the Fisher–Hartwig conjecture breaks down. However, this conjecture has been proved in all the cases in which it is suspected to apply [[#References|[a2]]], [[#References|[a5]]]. It is believed that the Basor–Tracy conjecture [[#References|[a4]]], which is proved so far (2000) only in special cases, gives the correct answer for all cases.
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where $\Omega = \sum _ { r = 1 } ^ { R } ( \alpha _ { r } ^ { 2 } - \beta _ { r } ^ { 2 } )$. An explicit, but more complicated expression is known for the constant $E$. It has turned out that in some cases the Fisher–Hartwig conjecture breaks down. However, this conjecture has been proved in all the cases in which it is suspected to apply [[#References|[a2]]], [[#References|[a5]]]. It is believed that the Basor–Tracy conjecture [[#References|[a4]]], which is proved so far (2000) only in special cases, gives the correct answer for all cases.
  
The continuous analogue of Toeplitz determinants are the determinants of truncated Wiener–Hopf operators (cf. also [[Wiener–Hopf operator|Wiener–Hopf operator]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064056.png" /> be a complex-valued function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064057.png" /> defined on the real axis, and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064058.png" /> the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064059.png" />. The [[Integral operator|integral operator]] defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064060.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064061.png" /> is called a truncated Wiener–Hopf operator and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064062.png" />. Under the above assumption, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064063.png" /> is a trace-class operator. The asymptotics of the operator determinants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064064.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064065.png" />, for certain classes of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064066.png" /> is described by the Akhiezer–Kac formula, which is the continuous pendant of the strong Szegö limit theorem. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064068.png" /> such that its Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064069.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064070.png" /> and
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The continuous analogue of Toeplitz determinants are the determinants of truncated Wiener–Hopf operators (cf. also [[Wiener–Hopf operator|Wiener–Hopf operator]]). Let $k$ be a complex-valued function in $L ^ { 1 } ( \mathbf{R} ) \cap L ^ { \infty } ( \mathbf{R} )$ defined on the real axis, and denote by $\hat{k}$ the [[Fourier transform|Fourier transform]] of $k$. The [[Integral operator|integral operator]] defined on $L ^ { 2 } [ 0 , \tau ]$ with kernel $\hat { k } ( x - y )$ is called a truncated Wiener–Hopf operator and denoted by $W _ { \tau } ( k )$. Under the above assumption, $W _ { \tau } ( k )$ is a trace-class operator. The asymptotics of the operator determinants of $I + W _ { \tau } ( k )$, as $\tau \rightarrow \infty$, for certain classes of functions $k$ is described by the Akhiezer–Kac formula, which is the continuous pendant of the strong Szegö limit theorem. Suppose $a = 1 + k = \operatorname { exp } ( s )$, where $s \in L ^ { 1 } ( \mathbf{R} ) \cap L ^ { \infty } ( \mathbf{R} )$ such that its Fourier transform $\stackrel \frown {s} $ belongs to $L ^ { 1 } ( \mathbf{R} )$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064071.png" /></td> </tr></table>
+
\begin{equation*} \int _ { - \infty } ^ { \infty } | t | | \hat{s} ( t ) | ^ { 2 } d t &lt; \infty. \end{equation*}
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064072.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { lim } _ { \tau \rightarrow \infty } \frac { \operatorname { det } ( I + W _ { \tau } ( k ) ) } { G ( a ) ^ { \tau } } = E ( a ), \end{equation*}
  
with the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064073.png" /> and
+
with the constants $G ( a ) = \operatorname { exp } ( \hat{s} ( 0 ) )$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064074.png" /></td> </tr></table>
+
\begin{equation*} E ( a ) = \operatorname { exp } \left( \int _ { 0 } ^ { \infty } t \hat{s} ( t ) \hat{s} ( - t ) d t \right) . \end{equation*}
  
 
There are many further results for Wiener–Hopf determinants which are quite similar to those of the discrete case [[#References|[a1]]], [[#References|[a2]]].
 
There are many further results for Wiener–Hopf determinants which are quite similar to those of the discrete case [[#References|[a1]]], [[#References|[a2]]].
Line 69: Line 77:
 
Finally, analogues of the Szegö limit theorem have also been established for multi-dimensional (i.e., multi-level) Toeplitz and Wiener–Hopf operators, for pseudo-differential operators, and in several abstract settings.
 
Finally, analogues of the Szegö limit theorem have also been established for multi-dimensional (i.e., multi-level) Toeplitz and Wiener–Hopf operators, for pseudo-differential operators, and in several abstract settings.
  
Another direction deals with the asymptotic distribution of the singular values of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130640/s13064075.png" />, their analogues and generalizations. Results of such a type are called Avram–Parter theorems [[#References|[a2]]].
+
Another direction deals with the asymptotic distribution of the singular values of the matrices $T _ { n } ( a )$, their analogues and generalizations. Results of such a type are called Avram–Parter theorems [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) {{MR|1086453}} {{MR|1071374}} {{ZBL|0732.47029}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Böttcher, B. Silbermann, "Introduction to large truncated Toeplitz matrices" , Springer (1998)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Böttcher, B. Silbermann, "Operator-valued Szegö-Widom limit theorems" , ''Oper. Theory Adv. Appl.'' , '''71''' , Birkhäuser (1994) pp. 33–53</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.L. Basor, C.A. Tracy, "The Fisher–Hartwig conjecture and generalizations" ''Phys. A'' , '''177''' (1991) pp. 167–173</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> T. Ehrhardt, "Toeplitz determinants with several Fisher–Hartwig singularities" ''PhD Thesis Techn. Univ. Chemnitz'' (1997)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M.E. Fisher, R.E. Hartwig, "Toeplitz determinants: Some applications, theorems and conjectures" ''Adv. Chem. Phys.'' , '''15''' (1968) pp. 333–353</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> G. Szegö, "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion" ''Math. Ann.'' , '''76''' (1915) pp. 490–503</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Szegö, "On certain Hermitian forms associated with the Fourier series of a positive function" ''Comm. Sém. Math. Univ. Lund'' (1952) pp. 228–238</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> N.L. Zamarashkin, E.E. Tyrtyshnikov, "Distribution of eigenvalues and singular numbers of Toeplitz matrices under weakened requirements of the generating function" ''Mat. Sb.'' , '''188''' (1997) pp. 83–92 (In Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Widom, "Asymptotic behavior of block Toeplitz matrices and determinants. II" ''Adv. Math.'' , '''21''' (1976) pp. 1–29</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> E.L. Basor, H. Widom, "On a Toeplitz determinant identity of Borodin and Okounov" ''Integral Eq. Oper. Th.'' , '''37''' : 4 (2000) pp. 397–401</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) {{MR|1086453}} {{MR|1071374}} {{ZBL|0732.47029}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> A. Böttcher, B. Silbermann, "Introduction to large truncated Toeplitz matrices" , Springer (1998)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> A. Böttcher, B. Silbermann, "Operator-valued Szegö-Widom limit theorems" , ''Oper. Theory Adv. Appl.'' , '''71''' , Birkhäuser (1994) pp. 33–53</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> E.L. Basor, C.A. Tracy, "The Fisher–Hartwig conjecture and generalizations" ''Phys. A'' , '''177''' (1991) pp. 167–173</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> T. Ehrhardt, "Toeplitz determinants with several Fisher–Hartwig singularities" ''PhD Thesis Techn. Univ. Chemnitz'' (1997)</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M.E. Fisher, R.E. Hartwig, "Toeplitz determinants: Some applications, theorems and conjectures" ''Adv. Chem. Phys.'' , '''15''' (1968) pp. 333–353</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> G. Szegö, "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion" ''Math. Ann.'' , '''76''' (1915) pp. 490–503</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> G. Szegö, "On certain Hermitian forms associated with the Fourier series of a positive function" ''Comm. Sém. Math. Univ. Lund'' (1952) pp. 228–238</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> N.L. Zamarashkin, E.E. Tyrtyshnikov, "Distribution of eigenvalues and singular numbers of Toeplitz matrices under weakened requirements of the generating function" ''Mat. Sb.'' , '''188''' (1997) pp. 83–92 (In Russian)</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> H. Widom, "Asymptotic behavior of block Toeplitz matrices and determinants. II" ''Adv. Math.'' , '''21''' (1976) pp. 1–29</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> E.L. Basor, H. Widom, "On a Toeplitz determinant identity of Borodin and Okounov" ''Integral Eq. Oper. Th.'' , '''37''' : 4 (2000) pp. 397–401</td></tr></table>

Revision as of 16:58, 1 July 2020

Let $a$ be a complex-valued function defined on the complex unit circle $\bf T$, with Fourier coefficients

\begin{equation*} a _ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } a ( e ^ { i \theta } ) e ^ { - i n \theta } d \theta. \end{equation*}

Szegö limit theorems describe the behaviour of the determinants of the Toeplitz matrices $T _ { n } ( a ) = ( a _ { j - k } ) _ { j , k = 0 } ^ { n - 1 }$, as $n$ tends to infinity, for certain classes of functions $a$ (cf. also Toeplitz matrix).

For real positive functions $a \in L ^ { 1 } ( {\bf T} )$ for which $\operatorname { log } a \in L ^ { 1 } (\bf T )$, G. Szegö [a7] has proved that

\begin{equation} \tag{a1} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { \operatorname { det } T _ { n - 1 } ( a ) } = G ( a ), \end{equation}

with the constant $G ( a ) = \operatorname { exp } ( [ \operatorname { log } a ] _ { 0 } )$. Here, $[ \operatorname { log } a ] _ { k }$ stands for the $k$th Fourier coefficient of the logarithm of $a$. A statement of type (a1) is referred to as a first Szegö limit theorem. Szegö's result has been considerably extended. In particular, (a1) holds for functions that are the exponentials of continuous complex-valued functions defined on the unit circle.

The strong Szegö limit theorem states that

\begin{equation} \tag{a2} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { G ( a ) ^ { N } } = E ( a ), \end{equation}

with the constant $E ( a )$ defined by

\begin{equation*} E ( a ) = \operatorname { exp } \left( \sum _ { k = 1 } ^ { \infty } k [ \operatorname { log } a ] _ { k } [ \operatorname { log } a ]_{ - k} \right). \end{equation*}

Relation (a2) was first proved by Szegö [a8] for positive real functions whose derivatives satisfy a Hölder–Lipschitz condition. This result has been generalized too. For instance, the strong Szegö limit theorem holds for functions that are the exponentials of continuous and sufficiently smooth complex-valued functions defined on the unit circle.

Such results about the asymptotics of Toeplitz determinants can be used to obtain information about the asymptotic distribution of the eigenvalues $\{ \lambda _ { k } ^ { ( n ) } \} _ { k = 1 } ^ { n }$ of the matrices $T _ { n } ( a )$. It turns out that

\begin{equation} \tag{a3} \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f ( \lambda _ { k } ^ { ( n ) } ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } f ( a ( e ^ { i \theta } ) ) d \theta + o ( 1 ), \end{equation}

as $n \rightarrow \infty$, if, for instance, one of the following assumptions is satisfied:

$a \in L ^ { 1 } ( {\bf T} )$ is real-valued and $f$ is a continuous function on the real line with a compact support [a9];

$a$ is a continuous complex-valued function and $f$ is an analytic function defined on an open neighbourhood of the set

\begin{equation*} \operatorname{spec} T ( a ) = \operatorname { Ran } ( a ) \bigcup \{ z \notin \operatorname { Ran } ( a ) : \text { wind } ( a - z ) \neq 0 \}. \end{equation*}

Here, $T ( a ) = ( a _ { j - k} ) _ { j , k = 0} ^ { \infty }$ stands for the Toeplitz operator acting on the Hilbert space $\text{l} ^ { 2 }$, $\operatorname {spec} T ( a )$ refers to its spectrum (cf. also Spectrum of an operator), $\operatorname{Ran}( a )$ stands for the range of the function $a$, and $\operatorname{wind}( a - z )$ denotes the winding number of the function $a ( e ^ { i \theta } ) - z$. The asymptotic formula (a3) is sometimes also called the first Szegö limit theorem or a first-order trace formula. A second-order trace formula, which is the pendant of the strong Szegö limit theorem, has also been established [a2], [a10].

Some work was also done in order to determine the higher-order terms of the asymptotic expansion of Toeplitz determinants [a1]. Exact formulas for Toeplitz determinants in terms of the Wiener–Hopf factorization (cf. also Wiener–Hopf method; Wiener–Hopf operator) of the generating function $a$ do also exist (see, e.g., [a11]).

H. Widom [a10] was the first to give a crystal clear proof of the strong Szegö limit theorem, by an elegant application of ideas from operator theory and thereby replacing earlier long-winded proofs. With his approach he was able to generalize this theorem to the case of matrix-valued functions. Under the assumption that $a$ is a sufficiently smooth matrix-valued function defined on the unit circle for which is the exponential of a continuous function, (a2) still holds, but with constants defined by $G ( a ) = \operatorname { exp } ( [ \operatorname { log } \operatorname { det } a ] _ { 0 } )$ and $E ( a ) = \operatorname { det } T ( a ) T ( a ^ { - 1 } )$. The last expression has to be understood as an operator determinant. In this connection, the identity plays an important role, where $H ( a ) = ( a _ { 1 + j + k} )_{ j,k = 0}^{\infty}$ is a Hankel operator and $\tilde { a } ( e ^ { i \theta } ) = a ( e ^ { - i \theta } )$. Note that for sufficiently smooth and invertible matrix functions $a$ the operator $H ( a ) H ( \tilde{a} ^ { - 1 } )$ is a trace-class operator (cf. also Nuclear operator). An explicit expression for $E ( a )$ is not known yet (as of 2000), apart from special cases related to the scalar situation. On the other hand, an operator-valued version of the strong Szegö limit theorem has been established [a3].

The asymptotic behaviour of Toeplitz determinants changes considerably if the function $a$ is discontinuous. If $a$ possesses zeros, poles, jumps, or certain oscillations, then the asymptotics is predicted by the Fisher–Hartwig conjecture or by the more general Basor–Tracy conjecture. Let

\begin{equation*} a ( e ^ { i \theta } ) = b ( e ^ { i \theta } ) \prod _ { r = 1 } ^ { R } \omega _ { \alpha _ { r } , \beta _ { r } } ( e ^ { i ( \theta - \theta _ { r } ) } ), \end{equation*}

where $\theta _ { 1 } , \dots , \theta _ { R } \in [ 0,2 \pi )$ are distinct points, $b$ is the exponential of a sufficiently smooth function and $\alpha_r$, $\beta _ { r }$ are complex parameters. The function $\omega _ { \alpha , \beta }$ is defined as

\begin{equation*} \omega _ { \alpha , \beta } ( e ^ { i \theta } ) = ( 2 - 2 \operatorname { cos } \theta ) ^ { \alpha } e ^ { i \beta ( \theta - \pi ) } , 0 < \theta < 2 \pi . \end{equation*}

Then the Fisher–Hartwig conjecture [a6] asserts that

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \frac { \operatorname { det } T _ { n } ( a ) } { G ( b ) ^ { n } n ^ { \Omega } } = E, \end{equation*}

where $\Omega = \sum _ { r = 1 } ^ { R } ( \alpha _ { r } ^ { 2 } - \beta _ { r } ^ { 2 } )$. An explicit, but more complicated expression is known for the constant $E$. It has turned out that in some cases the Fisher–Hartwig conjecture breaks down. However, this conjecture has been proved in all the cases in which it is suspected to apply [a2], [a5]. It is believed that the Basor–Tracy conjecture [a4], which is proved so far (2000) only in special cases, gives the correct answer for all cases.

The continuous analogue of Toeplitz determinants are the determinants of truncated Wiener–Hopf operators (cf. also Wiener–Hopf operator). Let $k$ be a complex-valued function in $L ^ { 1 } ( \mathbf{R} ) \cap L ^ { \infty } ( \mathbf{R} )$ defined on the real axis, and denote by $\hat{k}$ the Fourier transform of $k$. The integral operator defined on $L ^ { 2 } [ 0 , \tau ]$ with kernel $\hat { k } ( x - y )$ is called a truncated Wiener–Hopf operator and denoted by $W _ { \tau } ( k )$. Under the above assumption, $W _ { \tau } ( k )$ is a trace-class operator. The asymptotics of the operator determinants of $I + W _ { \tau } ( k )$, as $\tau \rightarrow \infty$, for certain classes of functions $k$ is described by the Akhiezer–Kac formula, which is the continuous pendant of the strong Szegö limit theorem. Suppose $a = 1 + k = \operatorname { exp } ( s )$, where $s \in L ^ { 1 } ( \mathbf{R} ) \cap L ^ { \infty } ( \mathbf{R} )$ such that its Fourier transform $\stackrel \frown {s} $ belongs to $L ^ { 1 } ( \mathbf{R} )$ and

\begin{equation*} \int _ { - \infty } ^ { \infty } | t | | \hat{s} ( t ) | ^ { 2 } d t < \infty. \end{equation*}

Then

\begin{equation*} \operatorname { lim } _ { \tau \rightarrow \infty } \frac { \operatorname { det } ( I + W _ { \tau } ( k ) ) } { G ( a ) ^ { \tau } } = E ( a ), \end{equation*}

with the constants $G ( a ) = \operatorname { exp } ( \hat{s} ( 0 ) )$ and

\begin{equation*} E ( a ) = \operatorname { exp } \left( \int _ { 0 } ^ { \infty } t \hat{s} ( t ) \hat{s} ( - t ) d t \right) . \end{equation*}

There are many further results for Wiener–Hopf determinants which are quite similar to those of the discrete case [a1], [a2].

Finally, analogues of the Szegö limit theorem have also been established for multi-dimensional (i.e., multi-level) Toeplitz and Wiener–Hopf operators, for pseudo-differential operators, and in several abstract settings.

Another direction deals with the asymptotic distribution of the singular values of the matrices $T _ { n } ( a )$, their analogues and generalizations. Results of such a type are called Avram–Parter theorems [a2].

References

[a1] A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) MR1086453 MR1071374 Zbl 0732.47029
[a2] A. Böttcher, B. Silbermann, "Introduction to large truncated Toeplitz matrices" , Springer (1998)
[a3] A. Böttcher, B. Silbermann, "Operator-valued Szegö-Widom limit theorems" , Oper. Theory Adv. Appl. , 71 , Birkhäuser (1994) pp. 33–53
[a4] E.L. Basor, C.A. Tracy, "The Fisher–Hartwig conjecture and generalizations" Phys. A , 177 (1991) pp. 167–173
[a5] T. Ehrhardt, "Toeplitz determinants with several Fisher–Hartwig singularities" PhD Thesis Techn. Univ. Chemnitz (1997)
[a6] M.E. Fisher, R.E. Hartwig, "Toeplitz determinants: Some applications, theorems and conjectures" Adv. Chem. Phys. , 15 (1968) pp. 333–353
[a7] G. Szegö, "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion" Math. Ann. , 76 (1915) pp. 490–503
[a8] G. Szegö, "On certain Hermitian forms associated with the Fourier series of a positive function" Comm. Sém. Math. Univ. Lund (1952) pp. 228–238
[a9] N.L. Zamarashkin, E.E. Tyrtyshnikov, "Distribution of eigenvalues and singular numbers of Toeplitz matrices under weakened requirements of the generating function" Mat. Sb. , 188 (1997) pp. 83–92 (In Russian)
[a10] H. Widom, "Asymptotic behavior of block Toeplitz matrices and determinants. II" Adv. Math. , 21 (1976) pp. 1–29
[a11] E.L. Basor, H. Widom, "On a Toeplitz determinant identity of Borodin and Okounov" Integral Eq. Oper. Th. , 37 : 4 (2000) pp. 397–401
How to Cite This Entry:
Szegö limit theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Szeg%C3%B6_limit_theorems&oldid=24575
This article was adapted from an original article by T. EhrhardtB. Silbermann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article