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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:
n1100101.png
 
$#A+1 = 54 n = 4
 
$#C+1 = 54 : ~/encyclopedia/old_files/data/N110/N.1100010 Nehari extension problem
 
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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|done}}
 
  
Let  $  \varphi _ {0} , \varphi _ {- 1 }  , \varphi _ {- 2 }  , \dots $
+
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.
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 (
 
  
induces a bounded [[Linear operator|linear operator]] $  \Phi $
+
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.
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 $
+
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:
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:
 
  
$$
+
<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>
\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>
\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/n11001031.png" /></td> </tr></table>
\left (
 
  
 
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 $  f \in {\mathcal W} $
+
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
of the suboptimal Nehari extension problem for the sequence $  \varphi _ {0} , \varphi _ {- 1 }  , \varphi _ {- 2 }  , \dots $
 
is of the form
 
  
$$ \tag{a1 }
+
<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>
f ( \lambda ) = ( \alpha ( \lambda ) h ( \lambda ) + \beta ( \lambda ) ) ( \gamma ( \lambda ) h ( \lambda ) + \delta ( \lambda ) ) ^ {- 1 } ,
 
$$
 
  
where $  \lambda \in \mathbf T $
+
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
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| $  H  ^  \infty  $
+
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]]]).
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 (
 
  
to a full infinite matrix such that the resulting operator on $  {\mathcal l}  ^ {2} $
+
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]]].
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>

Revision as of 14:32, 7 June 2020

Let be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all satisfying the following conditions:

i) the th Fourier coefficient of is equal to for each ;

ii) the norm constraint holds true. Here, is the norm of as an element of the Lebesgue function space and is the unit circle. Instead of condition ii) one may require , 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

induces a bounded linear operator on , the Hilbert space of all square-summable sequences, such that its operator norm is at most one, i.e., . The suboptimal version of the problem is solvable if and only if . If , either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If , 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 in the Wiener algebra , i.e., when one requires additionally that , can be described as follows. In this case, it is assumed that the given sequence is absolutely summable. Let . Then the operators and are boundedly invertible on , and one can build the following infinite column vectors:

Now, consider the functions

Then, each solution of the suboptimal Nehari extension problem for the sequence is of the form

(a1)

where and is an arbitrary element of the Wiener algebra such that for and the th Fourier coefficient of is zero for each . Moreover, (a1) gives a one-to-one correspondence between all such and all solutions . The central solution, i.e., the solution , which one obtains when the free parameter in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution of the suboptimal Nehari extension problem that maximizes the entropy integral

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 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,

to a full infinite matrix such that the resulting operator on 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