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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100801.png" /> be given complex numbers. The Carathéodory–Toeplitz extension problem is to find (if possible) a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100802.png" />, analytic on the open unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100803.png" /> (cf. also [[Analytic function|Analytic function]]), such that
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a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100805.png" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100806.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100807.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100808.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100809.png" />. The problem is solvable if and only if the [[Toeplitz matrix|Toeplitz matrix]]
+
Let  $  a _ {0} \dots a _ {p} $
 +
be given complex numbers. The Carathéodory–Toeplitz extension problem is to find (if possible) a function  $  g $,
 +
analytic on the open unit disc  $  | z | < 1 $(
 +
cf. also [[Analytic function|Analytic function]]), such 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/c/c110/c110080/c11008010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
a)  $  g ( z ) = a _ {0} + 2a _ {1} z + \dots + 2a _ {p} z  ^ {p} + O ( z ^ {p + 1 } ) $,
 +
$  | z | < 1 $;
  
is positive semi-definite, and its solution is unique if and only if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008011.png" /> is singular (cf. also [[Degenerate matrix|Degenerate matrix]]). The Carathéodory–Toeplitz extension problem can be restated as a [[Carathéodory–Schur extension problem|Carathéodory–Schur extension problem]]. The Levinson algorithm from filtering theory provides a recursive method to compute the solutions of the problem. For these and related results, see [[#References|[a1]]], Chapt. 2.
+
b)  $  { \mathop{\rm Re} } g ( z ) \geq 0 $
 +
for all  $  | z | < 1 $.
 +
Put  $  a _ {- j }  = {\overline{ {a _ {j} }}\; } $
 +
for  $  j = 1 \dots p $.
 +
The problem is solvable if and only if the [[Toeplitz matrix|Toeplitz matrix]]
  
Instead of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008012.png" /> satisfying a) and b), one may also seek functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008014.png" />, in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008015.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008016.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008017.png" />. (The Wiener algebra is defined as the [[Banach algebra|Banach algebra]] of complex-valued functions on the unit disc having a [[Fourier series|Fourier series]]
+
$$ \tag{a1 }
 +
\Gamma = \left (
  
<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/c/c110/c110080/c11008018.png" /></td> </tr></table>
+
\begin{array}{cccc}
 +
a _ {0}  &a _ {- 1 }  &\dots  &a _ {- p }  \\
 +
a _ {1}  &a _ {0}  &\dots  &a _ {- p + 1 }  \\
 +
\vdots  &\vdots  &\vdots  &\vdots  \\
 +
a _ {p}  &a _ {p -1 }  &\dots  &a _ {0}  \\
 +
\end{array}
  
using pointwise multiplication. The phrase  "Wiener algebra"  is also used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008019.png" /> with convolution as multiplication. There are also weighted versions; cf. [[#References|[a2]]].)
+
\right )
 +
$$
  
In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008020.png" /> satisfies conditions a) and b). The Wiener algebra version of the problem is of particular interest if the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008021.png" /> is required to be strictly positive on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008022.png" />. The latter version of the problem is solvable if and only if the Toeplitz matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008023.png" /> in (a1) is positive definite, and in that case there are infinitely many solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008024.png" />, given by
+
is positive semi-definite, and its solution is unique if and only if, in addition,  $  \Gamma $
 +
is singular (cf. also [[Degenerate matrix|Degenerate matrix]]). The Carathéodory–Toeplitz extension problem can be restated as a [[Carathéodory–Schur extension problem|Carathéodory–Schur extension problem]]. The Levinson algorithm from filtering theory provides a recursive method to compute the solutions of the problem. For these and related results, see [[#References|[a1]]], Chapt. 2.
  
<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/c/c110/c110080/c11008025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
Instead of functions  $  g $
 +
satisfying a) and b), one may also seek functions  $  f $,
 +
$  f ( \zeta ) = \sum _ {k = - \infty }  ^  \infty  f _ {k} \zeta  ^ {k} $,
 +
in the Wiener algebra  $  {\mathcal W} $
 +
with the property  $  f ( \zeta ) \geq 0 $
 +
for every  $  \zeta \in \mathbf T $.  
 +
(The Wiener algebra is defined as the [[Banach algebra|Banach algebra]] of complex-valued functions on the unit disc having a [[Fourier series|Fourier series]]
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008026.png" /> is an arbitrary function in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008027.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008028.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008029.png" />, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008031.png" /> are uniquely determined by the data in the following way:
+
$$
 +
f ( z ) = \sum _ {n = - \infty } ^  \infty  {a _ {n} z  ^ {n} } , \quad \sum _ {n = - \infty } ^  \infty  \left | {a _ {n} } \right | < \infty,
 +
$$
  
<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/c/c110/c110080/c11008032.png" /></td> </tr></table>
+
using [[pointwise multiplication]]. The phrase  "Wiener algebra" is also used for  $  L _ {1} ( \mathbf R ) $
 +
with convolution as multiplication. There are also weighted versions; cf. [[#References|[a2]]].)
  
<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/c/c110/c110080/c11008033.png" /></td> </tr></table>
+
In this case,  $  g ( \zeta ) = f _ {0} + 2 \sum _ {k = 0 }  ^  \infty  f _ {k} \zeta  ^ {k} $
 +
satisfies conditions a) and b). The Wiener algebra version of the problem is of particular interest if the solution  $  f $
 +
is required to be strictly positive on the unit circle  $  \mathbf T $.
 +
The latter version of the problem is solvable if and only if the Toeplitz matrix  $  \Gamma $
 +
in (a1) is positive definite, and in that case there are infinitely many solutions  $  f $,
 +
given by
 +
 
 +
$$ \tag{a2 }
 +
f ( \zeta ) = {
 +
\frac{1 - \left | {h ( \zeta ) } \right |  ^ {2} }{\left | {u ( \zeta ) + h ( \zeta ) v ( \zeta ) } \right |  ^ {2} }
 +
} .
 +
$$
 +
 
 +
Here,  $  h $
 +
is an arbitrary function in the Wiener algebra  $  {\mathcal W} $
 +
with  $  | {h ( \zeta ) } | < 1 $
 +
for every  $  \zeta \in \mathbf T $,
 +
and the functions  $  u $
 +
and  $  v $
 +
are uniquely determined by the data in the following way:
 +
 
 +
$$
 +
u ( \zeta ) = ( x _ {0} + \zeta x _ {1} + \dots + \zeta  ^ {p} x _ {p} ) x _ {0} ^ {- {1 / 2 } } ,
 +
$$
 +
 
 +
$$
 +
v ( \zeta ) = ( y _ {0} + \zeta ^ {-1 } y _ {-1 }  + \dots + \zeta ^ {- p } y _ {- p }  ) y _ {0} ^ {- {
 +
\frac{1}{2}
 +
} } ,
 +
$$
  
 
where
 
where
  
<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/c/c110/c110080/c11008034.png" /></td> </tr></table>
+
$$
 +
\left (
 +
 
 +
\begin{array}{c}
 +
x _ {0}  \\
 +
x _ {1}  \\
 +
\vdots  \\
 +
x _ {p}  \\
 +
\end{array}
 +
 
 +
\right ) = \Gamma ^ {- 1 } \left (
 +
 
 +
\begin{array}{c}
 +
1  \\
 +
0  \\
 +
\vdots  \\
 +
0  \\
 +
\end{array}
 +
 
 +
\right ) , \quad \left (
 +
 
 +
\begin{array}{c}
 +
y _ {- p }  \\
 +
\vdots  \\
 +
y _ {- 1 }  \\
 +
y _ {0}  \\
 +
\end{array}
 +
 
 +
\right ) = \Gamma ^ {- 1 } \left (
 +
 
 +
\begin{array}{c}
 +
y _ {0}  \\
 +
\vdots  \\
 +
0  \\
 +
1  \\
 +
\end{array}
 +
 
 +
\right ) .
 +
$$
  
The central solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008035.png" />, which appears when the free parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008036.png" /> in (a2) is set to zero, is the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008037.png" /> with the additional property that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008038.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008039.png" /> is equal to zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008040.png" />, and for this reason the central solution is also referred to as the band extension. The central solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008041.png" /> is also the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008042.png" /> that maximizes the entropy integral
+
The central solution $  f _ {\textrm{ cen  }  } ( \zeta ) = | {u ( \zeta ) } | ^ {-2 } $,  
 +
which appears when the free parameter $  h $
 +
in (a2) is set to zero, is the unique solution $  f $
 +
with the additional property that the $  j $
 +
th Fourier coefficient of $  f ^ {- 1 } $
 +
is equal to zero for $  | j | > p $,  
 +
and for this reason the central solution is also referred to as the band extension. The central solution $  f _ {\textrm{ cen  }  } $
 +
is also the unique solution $  f $
 +
that maximizes the entropy integral
  
<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/c/c110/c110080/c11008043.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{2 \pi }
 +
} \int\limits _ {- \pi } ^  \pi  { { \mathop{\rm log} } f ( e ^ {it } ) }  {dt } .
 +
$$
  
Proofs of the above results may derived by applying the [[Band method|band method]] (see [[#References|[a2]]], Sect. XXXV.3), which is a general scheme for dealing with a variety of positive and contractive (operator) extension problems from a unified point of view. (The word  "band"  refers to a decomposition of an algebra with involution, reminiscent of the use of bands as in the theory of [[Decomposition|decomposition]] or Riesz spaces (cf. [[Riesz space|Riesz space]]). It refers, in fact, to a  "band pattern" , i.e. a band in a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008044.png" />, cf. also [[Partially specified matrices, completion of|Partially specified matrices, completion of]].)
+
Proofs of the above results may derived by applying the [[Band method|band method]] (see [[#References|[a2]]], Sect. XXXV.3), which is a general scheme for dealing with a variety of positive and contractive (operator) extension problems from a unified point of view. (The word  "band"  refers to a decomposition of an algebra with involution, reminiscent of the use of bands as in the theory of [[Decomposition|decomposition]] or Riesz spaces (cf. [[Riesz space|Riesz space]]). It refers, in fact, to a  "band pattern" , i.e. a band in a matrix $  \{ {( i,j ) } : {| {i - j } | \leq  m } \} $,  
 +
cf. also [[Partially specified matrices, completion of|Partially specified matrices, completion of]].)
  
 
The Carathéodory–Toeplitz extension problem has natural generalizations for matrix- and operator-valued functions. The problem also has a continuous analogue (with the role of the open unit disc being replaced by the upper half-plane) and non-stationary versions for finite or infinite operator matrices.
 
The Carathéodory–Toeplitz extension problem has natural generalizations for matrix- and operator-valued functions. The problem also has a continuous analogue (with the role of the open unit disc being replaced by the upper half-plane) and non-stationary versions for finite or infinite operator matrices.

Latest revision as of 11:06, 30 May 2020


Let $ a _ {0} \dots a _ {p} $ be given complex numbers. The Carathéodory–Toeplitz extension problem is to find (if possible) a function $ g $, analytic on the open unit disc $ | z | < 1 $( cf. also Analytic function), such that

a) $ g ( z ) = a _ {0} + 2a _ {1} z + \dots + 2a _ {p} z ^ {p} + O ( z ^ {p + 1 } ) $, $ | z | < 1 $;

b) $ { \mathop{\rm Re} } g ( z ) \geq 0 $ for all $ | z | < 1 $. Put $ a _ {- j } = {\overline{ {a _ {j} }}\; } $ for $ j = 1 \dots p $. The problem is solvable if and only if the Toeplitz matrix

$$ \tag{a1 } \Gamma = \left ( \begin{array}{cccc} a _ {0} &a _ {- 1 } &\dots &a _ {- p } \\ a _ {1} &a _ {0} &\dots &a _ {- p + 1 } \\ \vdots &\vdots &\vdots &\vdots \\ a _ {p} &a _ {p -1 } &\dots &a _ {0} \\ \end{array} \right ) $$

is positive semi-definite, and its solution is unique if and only if, in addition, $ \Gamma $ is singular (cf. also Degenerate matrix). The Carathéodory–Toeplitz extension problem can be restated as a Carathéodory–Schur extension problem. The Levinson algorithm from filtering theory provides a recursive method to compute the solutions of the problem. For these and related results, see [a1], Chapt. 2.

Instead of functions $ g $ satisfying a) and b), one may also seek functions $ f $, $ f ( \zeta ) = \sum _ {k = - \infty } ^ \infty f _ {k} \zeta ^ {k} $, in the Wiener algebra $ {\mathcal W} $ with the property $ f ( \zeta ) \geq 0 $ for every $ \zeta \in \mathbf T $. (The Wiener algebra is defined as the Banach algebra of complex-valued functions on the unit disc having a Fourier series

$$ f ( z ) = \sum _ {n = - \infty } ^ \infty {a _ {n} z ^ {n} } , \quad \sum _ {n = - \infty } ^ \infty \left | {a _ {n} } \right | < \infty, $$

using pointwise multiplication. The phrase "Wiener algebra" is also used for $ L _ {1} ( \mathbf R ) $ with convolution as multiplication. There are also weighted versions; cf. [a2].)

In this case, $ g ( \zeta ) = f _ {0} + 2 \sum _ {k = 0 } ^ \infty f _ {k} \zeta ^ {k} $ satisfies conditions a) and b). The Wiener algebra version of the problem is of particular interest if the solution $ f $ is required to be strictly positive on the unit circle $ \mathbf T $. The latter version of the problem is solvable if and only if the Toeplitz matrix $ \Gamma $ in (a1) is positive definite, and in that case there are infinitely many solutions $ f $, given by

$$ \tag{a2 } f ( \zeta ) = { \frac{1 - \left | {h ( \zeta ) } \right | ^ {2} }{\left | {u ( \zeta ) + h ( \zeta ) v ( \zeta ) } \right | ^ {2} } } . $$

Here, $ h $ is an arbitrary function in the Wiener algebra $ {\mathcal W} $ with $ | {h ( \zeta ) } | < 1 $ for every $ \zeta \in \mathbf T $, and the functions $ u $ and $ v $ are uniquely determined by the data in the following way:

$$ u ( \zeta ) = ( x _ {0} + \zeta x _ {1} + \dots + \zeta ^ {p} x _ {p} ) x _ {0} ^ {- {1 / 2 } } , $$

$$ v ( \zeta ) = ( y _ {0} + \zeta ^ {-1 } y _ {-1 } + \dots + \zeta ^ {- p } y _ {- p } ) y _ {0} ^ {- { \frac{1}{2} } } , $$

where

$$ \left ( \begin{array}{c} x _ {0} \\ x _ {1} \\ \vdots \\ x _ {p} \\ \end{array} \right ) = \Gamma ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \\ \end{array} \right ) , \quad \left ( \begin{array}{c} y _ {- p } \\ \vdots \\ y _ {- 1 } \\ y _ {0} \\ \end{array} \right ) = \Gamma ^ {- 1 } \left ( \begin{array}{c} y _ {0} \\ \vdots \\ 0 \\ 1 \\ \end{array} \right ) . $$

The central solution $ f _ {\textrm{ cen } } ( \zeta ) = | {u ( \zeta ) } | ^ {-2 } $, which appears when the free parameter $ h $ in (a2) is set to zero, is the unique solution $ f $ with the additional property that the $ j $ th Fourier coefficient of $ f ^ {- 1 } $ is equal to zero for $ | j | > p $, and for this reason the central solution is also referred to as the band extension. The central solution $ f _ {\textrm{ cen } } $ is also the unique solution $ f $ that maximizes the entropy integral

$$ { \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ \pi { { \mathop{\rm log} } f ( e ^ {it } ) } {dt } . $$

Proofs of the above results may derived by applying the band method (see [a2], Sect. XXXV.3), which is a general scheme for dealing with a variety of positive and contractive (operator) extension problems from a unified point of view. (The word "band" refers to a decomposition of an algebra with involution, reminiscent of the use of bands as in the theory of decomposition or Riesz spaces (cf. Riesz space). It refers, in fact, to a "band pattern" , i.e. a band in a matrix $ \{ {( i,j ) } : {| {i - j } | \leq m } \} $, cf. also Partially specified matrices, completion of.)

The Carathéodory–Toeplitz extension problem has natural generalizations for matrix- and operator-valued functions. The problem also has a continuous analogue (with the role of the open unit disc being replaced by the upper half-plane) and non-stationary versions for finite or infinite operator matrices.

References

[a1] C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a2] I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators" , II , Birkhäuser (1993)
How to Cite This Entry:
Carathéodory-Toeplitz extension problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory-Toeplitz_extension_problem&oldid=23228
This article was adapted from an original article by I. GohbergM.A. Kaashoek (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article