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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100101.png" /> be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100102.png" /> satisfying the following conditions:
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i) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100103.png" />th Fourier coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100105.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100106.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100107.png" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
ii) the norm constraint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100108.png" /> holds true. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n1100109.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001010.png" /> as an element of the Lebesgue function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001012.png" /> is the unit circle. Instead of condition ii) one may require <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001013.png" />, and in the latter case one calls the problem suboptimal.
+
Let  $  \varphi _ {0} , \varphi _ {- 1 }  , \varphi _ {- 2 }  , \dots $
 +
be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all  $  f \in L _  \infty  ( \mathbf T ) $
 +
satisfying the following conditions:
 +
 
 +
i) the  $  n $
 +
th Fourier coefficient  $  c _ {n} ( f ) $
 +
of  $  f $
 +
is equal to  $  \varphi _ {n} $
 +
for each  $  n \leq  0 $;
 +
 
 +
ii) the norm constraint $  \| f \| _  \infty  \leq  1 $
 +
holds true. Here, $  \| f \| _  \infty  $
 +
is the norm of $  f $
 +
as an element of the Lebesgue function space $  L _  \infty  ( \mathbf T ) $
 +
and $  \mathbf T $
 +
is the unit circle. Instead of condition ii) one may require $  \| f \| _  \infty  < 1 $,  
 +
and in the latter case one calls the problem suboptimal.
  
 
The Nehari extension problem is not always solvable. In fact (see [[#References|[a12]]]), the problem has a solution if and only if the infinite [[Hankel matrix]]
 
The Nehari extension problem is not always solvable. In fact (see [[#References|[a12]]]), the problem has a solution if and only if the infinite [[Hankel matrix]]
  
<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/n/n110/n110010/n11001014.png" /></td> </tr></table>
+
$$
 +
\left (
 +
 
 +
\begin{array}{cccc}
 +
\varphi _ {0}  &\varphi _ {- 1 }  &\varphi _ {- 1 }  &\cdot  \\
 +
\varphi _ {- 1 }  &\varphi _ {- 2 }  &\varphi _ {- 3 }  &\cdot  \\
 +
\varphi _ {- 2 }  &\varphi _ {- 3 }  &\varphi _ {- 4 }  &\cdot  \\
 +
\cdot  &\cdot  &\cdot  &\cdot  \\
 +
\end{array}
 +
 
 +
\right )
 +
$$
 +
 
 +
induces a bounded [[Linear operator|linear operator]]  $  \Phi $
 +
on  $  {\mathcal l}  ^ {2} $,
 +
the [[Hilbert space|Hilbert space]] of all square-summable sequences, such that its operator norm is at most one, i.e.,  $  \| \Phi \| \leq  1 $.
 +
The suboptimal version of the problem is solvable if and only if  $  \| \Phi \| < 1 $.
 +
If  $  \| \Phi \| = 1 $,
 +
either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If  $  \| \Phi \| < 1 $,
 +
then the problem and its suboptimal version always have infinitely many solutions, which can be parametrized by a fractional-linear mapping.
 +
 
 +
For the suboptimal case, the set of all solutions  $  f $
 +
in the Wiener algebra  $  {\mathcal W} $,
 +
i.e., when one requires additionally that  $  \sum _ {n = - \infty }  ^  \infty  | {c _ {n} ( f ) } | < \infty $,
 +
can be described as follows. In this case, it is assumed that the given sequence  $  \varphi _ {0} , \varphi _ {- 1 }  , \varphi _ {- 2 }  , \dots $
 +
is absolutely summable. Let  $  \| \Phi \| < 1 $.
 +
Then the operators  $  I - \Phi  ^ {*} \Phi $
 +
and  $  I - \Phi \Phi  ^ {*} $
 +
are boundedly invertible on  $  {\mathcal l}  ^ {2} $,
 +
and one can build the following infinite column vectors:
 +
 
 +
$$
 +
\left (
 +
 
 +
\begin{array}{c}
 +
a _ {0}  \\
 +
a _ {- 1 }  \\
 +
a _ {- 2 }  \\
 +
\vdots  \\
 +
\end{array}
 +
 
 +
\right ) = ( I - \Phi \Phi  ^ {*} ) ^ {- 1 } \left (
 +
 
 +
\begin{array}{c}
 +
1  \\
 +
0  \\
 +
0  \\
 +
\vdots  \\
 +
\end{array}
 +
 
 +
\right ) ,
 +
$$
 +
 
 +
$$
 +
\left (
 +
 
 +
\begin{array}{c}
 +
c _ {0}  \\
 +
c _ {1}  \\
 +
c _ {2}  \\
 +
\vdots  \\
 +
\end{array}
 +
 
 +
\right ) = \Phi  ^ {*} \left (
 +
 
 +
\begin{array}{c}
 +
a _ {0}  \\
 +
a _ {- 1 }  \\
 +
a _ {- 2 }  \\
 +
\vdots  \\
 +
\end{array}
  
induces a bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001016.png" />, the [[Hilbert space|Hilbert space]] of all square-summable sequences, such that its operator norm is at most one, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001017.png" />. The suboptimal version of the problem is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001019.png" />, either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001020.png" />, then the problem and its suboptimal version always have infinitely many solutions, which can be parametrized by a fractional-linear mapping.
+
\right ) , \left (
  
For the suboptimal case, the set of all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001021.png" /> in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001022.png" />, i.e., when one requires additionally that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001023.png" />, can be described as follows. In this case, it is assumed that the given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001024.png" /> is absolutely summable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001025.png" />. Then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001027.png" /> are boundedly invertible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001028.png" />, and one can build the following infinite column vectors:
+
\begin{array}{c}
 +
d _ {0}  \\
 +
d _ {1}  \\
 +
d _ {2}  \\
 +
\vdots  \\
 +
\end{array}
  
<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/n/n110/n110010/n11001029.png" /></td> </tr></table>
+
\right ) = ( I - \Phi  ^ {*} \Phi ) ^ {- 1 } \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/n/n110/n110010/n11001030.png" /></td> </tr></table>
+
\begin{array}{c}
 +
1  \\
 +
0 \\
 +
0  \\
 +
\vdots  \\
 +
\end{array}
  
<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/n/n110/n110010/n11001031.png" /></td> </tr></table>
+
\right ) ,
 +
$$
 +
 
 +
$$
 +
\left (
 +
 
 +
\begin{array}{c}
 +
b _ {0}  \\
 +
b _ {- 1 }  \\
 +
b _ {- 2 }  \\
 +
\vdots  \\
 +
\end{array}
 +
 
 +
\right ) = \Phi \left (
 +
 
 +
\begin{array}{c}
 +
d _ {0}  \\
 +
d _ {1}  \\
 +
d _ {2}  \\
 +
\vdots  \\
 +
\end{array}
 +
 
 +
\right ) .
 +
$$
  
 
Now, consider the functions
 
Now, consider the functions
  
<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/n/n110/n110010/n11001032.png" /></td> </tr></table>
+
$$
 +
\alpha ( \lambda ) = \sum _ {j = - \infty } ^ { 0 }  a _ {j} a _ {0} ^ {- {1 / 2 } } \lambda  ^ {j} ,
 +
$$
  
<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/n/n110/n110010/n11001033.png" /></td> </tr></table>
+
$$
 +
\gamma ( \lambda ) = \sum _ {j = 0 } ^  \infty  c _ {j} a _ {0} ^ {- {1 / 2 } } \lambda  ^ {j} , \delta ( \lambda ) = \sum _ {j = 0 } ^  \infty  d _ {j} d _ {0} ^ {- {1 / 2 } } \lambda  ^ {j} ,
 +
$$
  
<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/n/n110/n110010/n11001034.png" /></td> </tr></table>
+
$$
 +
\beta ( \lambda ) = \sum _ {j = - \infty } ^ { 0 }  b _ {j} b _ {0} ^ {- {1 / 2 } } \lambda  ^ {j} .
 +
$$
  
Then, each solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001035.png" /> of the suboptimal Nehari extension problem for the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001036.png" /> is of the form
+
Then, each solution $  f \in {\mathcal W} $
 +
of the suboptimal Nehari extension problem for the sequence $  \varphi _ {0} , \varphi _ {- 1 }  , \varphi _ {- 2 }  , \dots $
 +
is of the form
  
<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/n/n110/n110010/n11001037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
f ( \lambda ) = ( \alpha ( \lambda ) h ( \lambda ) + \beta ( \lambda ) ) ( \gamma ( \lambda ) h ( \lambda ) + \delta ( \lambda ) ) ^ {- 1 } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001039.png" /> is an arbitrary element of the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001042.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001043.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001044.png" /> is zero for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001045.png" />. Moreover, (a1) gives a one-to-one correspondence between all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001046.png" /> and all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001047.png" />. The central solution, i.e., the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001048.png" />, which one obtains when the free parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001049.png" /> in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001050.png" /> of the suboptimal Nehari extension problem that maximizes the entropy integral
+
where $  \lambda \in \mathbf T $
 +
and $  h $
 +
is an arbitrary element of the Wiener algebra $  {\mathcal W} $
 +
such that $  | {h ( \lambda ) } | < 1 $
 +
for $  \lambda \in \mathbf T $
 +
and the n $
 +
th Fourier coefficient of $  h $
 +
is zero for each $  n \leq  0 $.  
 +
Moreover, (a1) gives a one-to-one correspondence between all such $  h $
 +
and all solutions $  f $.  
 +
The central solution, i.e., the solution $  f _ {\textrm{ cen  }  } ( \lambda ) = \beta ( \lambda ) \delta ( \lambda ) ^ {- 1 } $,  
 +
which one obtains when the free parameter $  h $
 +
in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution $  f \in {\mathcal W} $
 +
of the suboptimal Nehari extension problem 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/n/n110/n110010/n11001051.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{2 \pi }
 +
} \int\limits _ {- \pi } ^  \pi  { { \mathop{\rm log} } ( 1 - \left | {f ( e ^ {it } ) } \right |  ^ {2} ) }  {dt } .
 +
$$
  
The Nehari extension problem has natural generalizations for matrix-valued and operator-valued functions, and it has two-block and four-block analogues. In the matrix-valued case, a superoptimal Nehari extension problem is studied also. In the latter problem the constraint is made not only for the norm, but also for a number of first singular values [[#References|[a13]]]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [[#References|[a1]]]), the commutant-lifting approach (see [[#References|[a6]]] and [[Commutant lifting theorem|Commutant lifting theorem]]), the [[Band method|band method]] (see [[#References|[a10]]]), reproducing-kernel Hilbert space techniques (see [[#References|[a5]]]), and Beurling–Lax methods in Krein spaces (see [[#References|[a4]]] and [[Krein space|Krein space]]). The results are used in [[H^infinity-control-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001052.png" /> control theory]] (see [[#References|[a8]]]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional representation (a1) can be represented explicitly in state-space form (see [[#References|[a9]]] and [[#References|[a3]]]).
+
The Nehari extension problem has natural generalizations for matrix-valued and operator-valued functions, and it has two-block and four-block analogues. In the matrix-valued case, a superoptimal Nehari extension problem is studied also. In the latter problem the constraint is made not only for the norm, but also for a number of first singular values [[#References|[a13]]]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [[#References|[a1]]]), the commutant-lifting approach (see [[#References|[a6]]] and [[Commutant lifting theorem|Commutant lifting theorem]]), the [[Band method|band method]] (see [[#References|[a10]]]), reproducing-kernel Hilbert space techniques (see [[#References|[a5]]]), and Beurling–Lax methods in Krein spaces (see [[#References|[a4]]] and [[Krein space|Krein space]]). The results are used in [[H^infinity-control-theory| $  H  ^  \infty  $
 +
control theory]] (see [[#References|[a8]]]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional representation (a1) can be represented explicitly in state-space form (see [[#References|[a9]]] and [[#References|[a3]]]).
  
 
The Nehari extension problem also has non-stationary versions, in which the role of analytic functions is taken over by lower-triangular matrices. An example is the problem to complete a given lower-triangular array of numbers,
 
The Nehari extension problem also has non-stationary versions, in which the role of analytic functions is taken over by lower-triangular matrices. An example is the problem to complete a given lower-triangular array of numbers,
  
<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/n/n110/n110010/n11001053.png" /></td> </tr></table>
+
$$
 +
\left (
 +
 
 +
\begin{array}{cccc}
 +
\varphi _ {00 }  &{}  &{}  &{}  \\
 +
\varphi _ {10 }  &\varphi _ {11 }  &{}  &{}  \\
 +
\varphi _ {20 }  &\varphi _ {21 }  &\varphi _ {22 }  &{}  \\
 +
\cdot  &\cdot  &\cdot  &\cdot  \\
 +
\end{array}
 +
 
 +
\right ) ,
 +
$$
  
to a full infinite matrix such that the resulting operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001054.png" /> is bounded and has operator norm at most one. The non-stationary variants of the Nehari extension problem have been treated in terms of nest algebras [[#References|[a2]]]. The main results for the stationary case carry over to the non-stationary case [[#References|[a11]]], [[#References|[a7]]].
+
to a full infinite matrix such that the resulting operator on $  {\mathcal l}  ^ {2} $
 +
is bounded and has operator norm at most one. The non-stationary variants of the Nehari extension problem have been treated in terms of nest algebras [[#References|[a2]]]. The main results for the stationary case carry over to the non-stationary case [[#References|[a11]]], [[#References|[a7]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Adamjan,  D.Z. Arov,  M.G. Krein,  "Infinite Hankel block matrices and related extension problems"  ''Transl. Amer. Math. Soc.'' , '''111'''  (1978)  pp. 133–156  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''6'''  (1971)  pp. 87–112</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.B. Arveson,  "Interpolation in nest algebras"  ''J. Funct. Anal.'' , '''20'''  (1975)  pp. 208–233</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Ball,  I. Gohberg,  L. Rodman,  "Interpolation of rational matrix functions" , ''Operator Theory: Advances and Applications'' , '''45''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Ball,  J.W. Helton,  "A Beurling–Lax theorem for Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001055.png" /> which contains classical interpolation theory"  ''J. Operator Th.'' , '''9'''  (1983)  pp. 107–142</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Dym,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001056.png" /> contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , ''CBMS'' , '''71''' , Amer. Math. Soc.  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Foias,  A.E. Frazho,  "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C. Foias,  A.E. Frazho,  I. Gohberg,  M.A. Kaashoek,  "Discrete time-variant interpolation as classical interpolation with an operator argument"  ''Integral Eq. Operator Th.'' , '''26'''  (1996)  pp. 371–403</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.A. Francis,  "A course in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001057.png" /> control theory" , Springer  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  K. Glover,  "All optimal Hankel-norm approximations of linear multivariable systems and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001058.png" />-error bounds"  ''Int. J. Control'' , '''39'''  (1984)  pp. 1115–1193</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Gohberg,  S. Goldberg,  M.A Kaashoek,  "Classes of linear operators II" , ''Operator Theory: Advances and Applications'' , '''63''' , Birkhäuser  (1993)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  I. Gohberg,  M.A. Kaashoek,  H.J. Woerdeman,  "The band method for positive and contractive extension problems: An alternative version and new applications"  ''Integral Eq. Operator Th.'' , '''12'''  (1989)  pp. 343–382</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  Z. Nehari,  "On bounded bilinear forms"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 153–162</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.V. Peller,  N.J. Young,  "Superoptimal analytic approximations of matrix functions"  ''J. Funct. Anal.'' , '''120'''  (1994)  pp. 300–343</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Adamjan,  D.Z. Arov,  M.G. Krein,  "Infinite Hankel block matrices and related extension problems"  ''Transl. Amer. Math. Soc.'' , '''111'''  (1978)  pp. 133–156  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''6'''  (1971)  pp. 87–112</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.B. Arveson,  "Interpolation in nest algebras"  ''J. Funct. Anal.'' , '''20'''  (1975)  pp. 208–233</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Ball,  I. Gohberg,  L. Rodman,  "Interpolation of rational matrix functions" , ''Operator Theory: Advances and Applications'' , '''45''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Ball,  J.W. Helton,  "A Beurling–Lax theorem for Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001055.png" /> which contains classical interpolation theory"  ''J. Operator Th.'' , '''9'''  (1983)  pp. 107–142</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Dym,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001056.png" /> contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , ''CBMS'' , '''71''' , Amer. Math. Soc.  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Foias,  A.E. Frazho,  "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C. Foias,  A.E. Frazho,  I. Gohberg,  M.A. Kaashoek,  "Discrete time-variant interpolation as classical interpolation with an operator argument"  ''Integral Eq. Operator Th.'' , '''26'''  (1996)  pp. 371–403</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.A. Francis,  "A course in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001057.png" /> control theory" , Springer  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  K. Glover,  "All optimal Hankel-norm approximations of linear multivariable systems and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110010/n11001058.png" />-error bounds"  ''Int. J. Control'' , '''39'''  (1984)  pp. 1115–1193</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Gohberg,  S. Goldberg,  M.A Kaashoek,  "Classes of linear operators II" , ''Operator Theory: Advances and Applications'' , '''63''' , Birkhäuser  (1993)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  I. Gohberg,  M.A. Kaashoek,  H.J. Woerdeman,  "The band method for positive and contractive extension problems: An alternative version and new applications"  ''Integral Eq. Operator Th.'' , '''12'''  (1989)  pp. 343–382</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  Z. Nehari,  "On bounded bilinear forms"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 153–162</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.V. Peller,  N.J. Young,  "Superoptimal analytic approximations of matrix functions"  ''J. Funct. Anal.'' , '''120'''  (1994)  pp. 300–343</TD></TR></table>

Latest revision as of 14:32, 7 June 2020


Let $ \varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots $ be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all $ f \in L _ \infty ( \mathbf T ) $ satisfying the following conditions:

i) the $ n $ th Fourier coefficient $ c _ {n} ( f ) $ of $ f $ is equal to $ \varphi _ {n} $ for each $ n \leq 0 $;

ii) the norm constraint $ \| f \| _ \infty \leq 1 $ holds true. Here, $ \| f \| _ \infty $ is the norm of $ f $ as an element of the Lebesgue function space $ L _ \infty ( \mathbf T ) $ and $ \mathbf T $ is the unit circle. Instead of condition ii) one may require $ \| f \| _ \infty < 1 $, and in the latter case one calls the problem suboptimal.

The Nehari extension problem is not always solvable. In fact (see [a12]), the problem has a solution if and only if the infinite Hankel matrix

$$ \left ( \begin{array}{cccc} \varphi _ {0} &\varphi _ {- 1 } &\varphi _ {- 1 } &\cdot \\ \varphi _ {- 1 } &\varphi _ {- 2 } &\varphi _ {- 3 } &\cdot \\ \varphi _ {- 2 } &\varphi _ {- 3 } &\varphi _ {- 4 } &\cdot \\ \cdot &\cdot &\cdot &\cdot \\ \end{array} \right ) $$

induces a bounded linear operator $ \Phi $ on $ {\mathcal l} ^ {2} $, the Hilbert space of all square-summable sequences, such that its operator norm is at most one, i.e., $ \| \Phi \| \leq 1 $. The suboptimal version of the problem is solvable if and only if $ \| \Phi \| < 1 $. If $ \| \Phi \| = 1 $, either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If $ \| \Phi \| < 1 $, then the problem and its suboptimal version always have infinitely many solutions, which can be parametrized by a fractional-linear mapping.

For the suboptimal case, the set of all solutions $ f $ in the Wiener algebra $ {\mathcal W} $, i.e., when one requires additionally that $ \sum _ {n = - \infty } ^ \infty | {c _ {n} ( f ) } | < \infty $, can be described as follows. In this case, it is assumed that the given sequence $ \varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots $ is absolutely summable. Let $ \| \Phi \| < 1 $. Then the operators $ I - \Phi ^ {*} \Phi $ and $ I - \Phi \Phi ^ {*} $ are boundedly invertible on $ {\mathcal l} ^ {2} $, and one can build the following infinite column vectors:

$$ \left ( \begin{array}{c} a _ {0} \\ a _ {- 1 } \\ a _ {- 2 } \\ \vdots \\ \end{array} \right ) = ( I - \Phi \Phi ^ {*} ) ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ \vdots \\ \end{array} \right ) , $$

$$ \left ( \begin{array}{c} c _ {0} \\ c _ {1} \\ c _ {2} \\ \vdots \\ \end{array} \right ) = \Phi ^ {*} \left ( \begin{array}{c} a _ {0} \\ a _ {- 1 } \\ a _ {- 2 } \\ \vdots \\ \end{array} \right ) , \left ( \begin{array}{c} d _ {0} \\ d _ {1} \\ d _ {2} \\ \vdots \\ \end{array} \right ) = ( I - \Phi ^ {*} \Phi ) ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ \vdots \\ \end{array} \right ) , $$

$$ \left ( \begin{array}{c} b _ {0} \\ b _ {- 1 } \\ b _ {- 2 } \\ \vdots \\ \end{array} \right ) = \Phi \left ( \begin{array}{c} d _ {0} \\ d _ {1} \\ d _ {2} \\ \vdots \\ \end{array} \right ) . $$

Now, consider the functions

$$ \alpha ( \lambda ) = \sum _ {j = - \infty } ^ { 0 } a _ {j} a _ {0} ^ {- {1 / 2 } } \lambda ^ {j} , $$

$$ \gamma ( \lambda ) = \sum _ {j = 0 } ^ \infty c _ {j} a _ {0} ^ {- {1 / 2 } } \lambda ^ {j} , \delta ( \lambda ) = \sum _ {j = 0 } ^ \infty d _ {j} d _ {0} ^ {- {1 / 2 } } \lambda ^ {j} , $$

$$ \beta ( \lambda ) = \sum _ {j = - \infty } ^ { 0 } b _ {j} b _ {0} ^ {- {1 / 2 } } \lambda ^ {j} . $$

Then, each solution $ f \in {\mathcal W} $ of the suboptimal Nehari extension problem for the sequence $ \varphi _ {0} , \varphi _ {- 1 } , \varphi _ {- 2 } , \dots $ is of the form

$$ \tag{a1 } f ( \lambda ) = ( \alpha ( \lambda ) h ( \lambda ) + \beta ( \lambda ) ) ( \gamma ( \lambda ) h ( \lambda ) + \delta ( \lambda ) ) ^ {- 1 } , $$

where $ \lambda \in \mathbf T $ and $ h $ is an arbitrary element of the Wiener algebra $ {\mathcal W} $ such that $ | {h ( \lambda ) } | < 1 $ for $ \lambda \in \mathbf T $ and the $ n $ th Fourier coefficient of $ h $ is zero for each $ n \leq 0 $. Moreover, (a1) gives a one-to-one correspondence between all such $ h $ and all solutions $ f $. The central solution, i.e., the solution $ f _ {\textrm{ cen } } ( \lambda ) = \beta ( \lambda ) \delta ( \lambda ) ^ {- 1 } $, which one obtains when the free parameter $ h $ in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution $ f \in {\mathcal W} $ of the suboptimal Nehari extension problem that maximizes the entropy integral

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

The Nehari extension problem has natural generalizations for matrix-valued and operator-valued functions, and it has two-block and four-block analogues. In the matrix-valued case, a superoptimal Nehari extension problem is studied also. In the latter problem the constraint is made not only for the norm, but also for a number of first singular values [a13]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [a1]), the commutant-lifting approach (see [a6] and Commutant lifting theorem), the band method (see [a10]), reproducing-kernel Hilbert space techniques (see [a5]), and Beurling–Lax methods in Krein spaces (see [a4] and Krein space). The results are used in $ H ^ \infty $ control theory (see [a8]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional representation (a1) can be represented explicitly in state-space form (see [a9] and [a3]).

The Nehari extension problem also has non-stationary versions, in which the role of analytic functions is taken over by lower-triangular matrices. An example is the problem to complete a given lower-triangular array of numbers,

$$ \left ( \begin{array}{cccc} \varphi _ {00 } &{} &{} &{} \\ \varphi _ {10 } &\varphi _ {11 } &{} &{} \\ \varphi _ {20 } &\varphi _ {21 } &\varphi _ {22 } &{} \\ \cdot &\cdot &\cdot &\cdot \\ \end{array} \right ) , $$

to a full infinite matrix such that the resulting operator on $ {\mathcal l} ^ {2} $ is bounded and has operator norm at most one. The non-stationary variants of the Nehari extension problem have been treated in terms of nest algebras [a2]. The main results for the stationary case carry over to the non-stationary case [a11], [a7].

References

[a1] V.M. Adamjan, D.Z. Arov, M.G. Krein, "Infinite Hankel block matrices and related extension problems" Transl. Amer. Math. Soc. , 111 (1978) pp. 133–156 Izv. Akad. Nauk SSSR Ser. Mat. , 6 (1971) pp. 87–112
[a2] W.B. Arveson, "Interpolation in nest algebras" J. Funct. Anal. , 20 (1975) pp. 208–233
[a3] J. Ball, I. Gohberg, L. Rodman, "Interpolation of rational matrix functions" , Operator Theory: Advances and Applications , 45 , Birkhäuser (1990)
[a4] J. Ball, J.W. Helton, "A Beurling–Lax theorem for Lie group which contains classical interpolation theory" J. Operator Th. , 9 (1983) pp. 107–142
[a5] H. Dym, " contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , CBMS , 71 , Amer. Math. Soc. (1989)
[a6] C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a7] C. Foias, A.E. Frazho, I. Gohberg, M.A. Kaashoek, "Discrete time-variant interpolation as classical interpolation with an operator argument" Integral Eq. Operator Th. , 26 (1996) pp. 371–403
[a8] B.A. Francis, "A course in control theory" , Springer (1987)
[a9] K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and the -error bounds" Int. J. Control , 39 (1984) pp. 1115–1193
[a10] I. Gohberg, S. Goldberg, M.A Kaashoek, "Classes of linear operators II" , Operator Theory: Advances and Applications , 63 , Birkhäuser (1993)
[a11] I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "The band method for positive and contractive extension problems: An alternative version and new applications" Integral Eq. Operator Th. , 12 (1989) pp. 343–382
[a12] Z. Nehari, "On bounded bilinear forms" Ann. of Math. , 65 (1957) pp. 153–162
[a13] V.V. Peller, N.J. Young, "Superoptimal analytic approximations of matrix functions" J. Funct. Anal. , 120 (1994) pp. 300–343
How to Cite This Entry:
Nehari extension problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nehari_extension_problem&oldid=49318
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