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Given a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664801.png" /> of analytic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664802.png" /> of the complex plane (or, in a more general context, of a Riemann surface), to find necessary and sufficient conditions for the solvability in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664803.png" /> of the interpolation problem
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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/n/n066/n066480/n0664804.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664805.png" /> is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664807.png" /> is some set of complex numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664808.png" /> usually runs through a countable (sometimes finite, sometimes even uncountable) index set. The classical result of G. Pick [[#References|[1]]] and R. Nevanlinna [[#References|[2]]] (for finite and countable subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n0664809.png" />, respectively) yields the solution of this problem, for example, in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648010.png" /> of analytic functions in the unit disc that are bounded by 1 in absolute value. The desired condition here is the non-negativity of the quadratic forms
+
Given a class  $  \mathfrak H $
 +
of analytic functions in a domain  $  G $
 +
of the complex plane (or, in a more general context, of a Riemann surface), to find necessary and sufficient conditions for the solvability in $  \mathfrak H $
 +
of the interpolation problem
  
<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/n066/n066480/n06648011.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
f ( z _  \alpha  )  = w _  \alpha  ,
 +
$$
  
The first proof of this proposition (see [[#References|[3]]], [[#References|[4]]]), as well as quite analogous and similar results for various other function classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648012.png" />, relied on algebraic and functional-theoretic methods. Later proofs, based, for example, on reducing the Nevanlinna–Pick problem to a moment problem or obtained from the point of view of the theory of Hilbert spaces, have made it possible to extend the result to uncountable subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648013.png" /> and pointed the way to possible generalizations (see [[#References|[3]]]–[[#References|[5]]]).
+
where  $  \{ z _  \alpha  \} $
 +
is a subset of  $  G $,
 +
$  \{ w _  \alpha  \} $
 +
is some set of complex numbers, and  $  \alpha $
 +
usually runs through a countable (sometimes finite, sometimes even uncountable) index set. The classical result of G. Pick [[#References|[1]]] and R. Nevanlinna [[#References|[2]]] (for finite and countable subsets  $  \{ z _  \alpha  \} \subset  G $,  
 +
respectively) yields the solution of this problem, for example, in the class  $  B _ {1} $
 +
of analytic functions in the unit disc that are bounded by 1 in absolute value. The desired condition here is the non-negativity of the quadratic forms
  
A natural development of the Nevanlinna–Pick problem, which necessitated an appeal to functional-analytic methods of investigation, was the question of the solvability of the interpolation problem (1) on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648014.png" /> of right-hand sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648015.png" />; in this case, as a rule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648016.png" /> is a countable set (a sequence) of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648017.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648018.png" /> may be one of various spaces of sequences of complex numbers. In connection with the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648019.png" /> of bounded analytic functions in the unit disc and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648020.png" /> of bounded sequences, a complete description of the corresponding point sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648021.png" /> (known as universal interpolation sequences) has been obtained (see [[#References|[6]]]) in the form of the condition
+
$$
 +
\sum _ {j, k = 1 } ^ { n }
  
<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/n066/n066480/n06648022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{1 - w _ {j} \overline{ {w _ {k} }}\; }{1 - z _ {j} \overline{ {z _ {k} }}\; }
  
This result played an important role in describing the structure of the maximal ideal space of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648023.png" /> (see [[#References|[7]]]) and was at the same time a starting point for extensive research into the Nevanlinna–Pick problem (in the above generalized formulation) for the [[Hardy classes|Hardy classes]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648024.png" /> and the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648025.png" /> (including weight spaces). It turned out that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648026.png" /> the solution is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648027.png" /> and is given by condition (2), while when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648028.png" /> it necessarily varies when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648030.png" /> are changed (see ). Another generalization of the Nevanlinna–Pick problem is connected with the interpolation problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648032.png" /> is some system of functionals in a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648033.png" />. The problem of describing the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648034.png" /> may be regarded as a generalization of the well-known [[Coefficient problem|coefficient problem]] for classes of analytic functions.
+
\xi _ {j} \overline{ {\xi _ {k} }}\; ,\ \
 +
n \in \mathbf N ,\ \
 +
\xi _ {j} \in \mathbf C .
 +
$$
 +
 
 +
The first proof of this proposition (see [[#References|[3]]], [[#References|[4]]]), as well as quite analogous and similar results for various other function classes  $  \mathfrak H $,
 +
relied on algebraic and functional-theoretic methods. Later proofs, based, for example, on reducing the Nevanlinna–Pick problem to a moment problem or obtained from the point of view of the theory of Hilbert spaces, have made it possible to extend the result to uncountable subsets  $  \{ z _  \alpha  \} \subset  G $
 +
and pointed the way to possible generalizations (see [[#References|[3]]]–[[#References|[5]]]).
 +
 
 +
A natural development of the Nevanlinna–Pick problem, which necessitated an appeal to functional-analytic methods of investigation, was the question of the solvability of the interpolation problem (1) on a class  $  W $
 +
of right-hand sides  $  \{ w _  \alpha  \} $;
 +
in this case, as a rule,  $  \{ z _  \alpha  \} $
 +
is a countable set (a sequence) of points of  $  G $,
 +
while  $  W $
 +
may be one of various spaces of sequences of complex numbers. In connection with the class  $  H  ^  \infty  $
 +
of bounded analytic functions in the unit disc and the space  $  l _  \infty  $
 +
of bounded sequences, a complete description of the corresponding point sequences  $  \{ z _  \alpha  \} $(
 +
known as universal interpolation sequences) has been obtained (see [[#References|[6]]]) in the form of the condition
 +
 
 +
$$ \tag{2 }
 +
\prod _ {\begin{array}{c}
 +
j = 1 \\
 +
j \neq k
 +
\end{array}
 +
} ^  \infty 
 +
\left |
 +
\frac{z _ {j} - z _ {k} }{1 - z _ {j} \overline{ {z _ {k} }}\; }
 +
\right |
 +
\geq  \delta  >  0,\ \
 +
k \in \mathbf N .
 +
$$
 +
 
 +
This result played an important role in describing the structure of the maximal ideal space of the algebra $  H  ^  \infty  $(
 +
see [[#References|[7]]]) and was at the same time a starting point for extensive research into the Nevanlinna–Pick problem (in the above generalized formulation) for the [[Hardy classes|Hardy classes]] $  H  ^ {q} $
 +
and the spaces $  l _ {p} $(
 +
including weight spaces). It turned out that when $  q = \infty $
 +
the solution is independent of $  p $
 +
and is given by condition (2), while when $  q < \infty $
 +
it necessarily varies when $  q $
 +
and $  p $
 +
are changed (see ). Another generalization of the Nevanlinna–Pick problem is connected with the interpolation problem $  \phi _  \alpha  ( f  ) = w _  \alpha  $,  
 +
where $  \{ \phi _  \alpha  \} $
 +
is some system of functionals in a class $  \mathfrak H $.  
 +
The problem of describing the set $  \{ \{ \phi _  \alpha  ( f  ) \} : f \in \mathfrak H \} $
 +
may be regarded as a generalization of the well-known [[Coefficient problem|coefficient problem]] for classes of analytic functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pick,  "Ueber die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden"  ''Math. Ann.'' , '''77'''  (1916)  pp. 7–23</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanlinna,  "Ueber beschränkte analytische Funktionen"  ''Ann. Acad. Sci. Fenn. Ser. A'' , '''32''' :  7  (1929)  pp. 1–15</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.G. Krein,  A.A. Nudel'man,  "The Markov moment problem and extremal problems" , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Sz.-Nagy,  A. Korányi,  "Rélations d'un problem de Nevanlinna et Pick avec la théorie des opérateurs de l'espace Hilbertien"  ''Acta Math. Acad. Sci. Hung.'' , '''7'''  (1956)  pp. 295–303</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Carleson,  "An interpolation problem for bounded analytic functions"  ''Amer. J. Math.'' , '''80''' :  4  (1958)  pp. 921–930</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.V. Shvedenko,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648035.png" />-interpolation sequences in the unit disk"  ''Math. USSR Sb.'' , '''46''' :  4  (1983)  pp. 473–492  ''Mat. Sb.'' , '''118''' :  4  (1982)  pp. 470–489</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pick,  "Ueber die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden"  ''Math. Ann.'' , '''77'''  (1916)  pp. 7–23</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanlinna,  "Ueber beschränkte analytische Funktionen"  ''Ann. Acad. Sci. Fenn. Ser. A'' , '''32''' :  7  (1929)  pp. 1–15</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.G. Krein,  A.A. Nudel'man,  "The Markov moment problem and extremal problems" , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Sz.-Nagy,  A. Korányi,  "Rélations d'un problem de Nevanlinna et Pick avec la théorie des opérateurs de l'espace Hilbertien"  ''Acta Math. Acad. Sci. Hung.'' , '''7'''  (1956)  pp. 295–303</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Carleson,  "An interpolation problem for bounded analytic functions"  ''Amer. J. Math.'' , '''80''' :  4  (1958)  pp. 921–930</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.V. Shvedenko,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648035.png" />-interpolation sequences in the unit disk"  ''Math. USSR Sb.'' , '''46''' :  4  (1983)  pp. 473–492  ''Mat. Sb.'' , '''118''' :  4  (1982)  pp. 470–489</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
This classical topic was reactivated and extended to matrix-valued functions in the 1960s and the beginning of the 1970s. Connections with operator theory became essential in the new developments (see, e.g., [[#References|[a1]]]–[[#References|[a3]]]). Applications to problems in control theory, which appeared in the 1980s, required a revision of the theory and the development of computational methods, in particular, for rational matrix functions (see [[#References|[a4]]], [[#References|[a5]]]).
 
This classical topic was reactivated and extended to matrix-valued functions in the 1960s and the beginning of the 1970s. Connections with operator theory became essential in the new developments (see, e.g., [[#References|[a1]]]–[[#References|[a3]]]). Applications to problems in control theory, which appeared in the 1980s, required a revision of the theory and the development of computational methods, in particular, for rational matrix functions (see [[#References|[a4]]], [[#References|[a5]]]).
  
See also [[H^infinity-control-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648036.png" /> control theory]].
+
See also [[H^infinity-control-theory| $  H  ^  \infty  $
 +
control theory]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. [V.M. Adamyan] Adamjan,  D.Z. Arov,  M.G. Krein,  "Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Tagaki problem"  ''Math. USSR Sb.'' , '''15'''  (1971)  pp. 31–73  ''Mat. Sb.'' , '''86''' :  1  (1971)  pp. 34–75</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Rosenblum,  J. Rovnyak,  "Hardy classes and operator theory" , Oxford Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sarason,  "Operator-theoretic aspects of the Nevanlinna–Pick interpolation problem"  S.C. Power (ed.) , ''Operators and function theory'' , Reidel  (1984)  pp. 279–314</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Kimura,  "Directional interpolation approach in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648037.png" />-optimization"  ''IEEE Trans. Autom. Control'' , '''32'''  (1987)  pp. 1085–1093</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.J.N. Limebeer,  B.D.O. Anderson,  "An interpolation theory approach to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648038.png" /> controller degree bounds"  ''Linear Algebra Appl.'' , '''98'''  (1988)  pp. 347–386</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.A. Ball,  "Nevanlinna–Pick interpolation: Generalizations and applications"  J.B. Conway (ed.) , ''Proc. Asymmetric Algebras and Invariant Subspaces. Conf. Indian Univ.'' , Pitman  (To appear)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Dym,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648039.png" /> contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , Amer. Math. Soc.  (1989)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.W. Helton,  "Operator theory, analytic functions, matrices, and electrical engineering" , Amer. Math. Soc.  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P.L. Duren,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648040.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. [V.M. Adamyan] Adamjan,  D.Z. Arov,  M.G. Krein,  "Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Tagaki problem"  ''Math. USSR Sb.'' , '''15'''  (1971)  pp. 31–73  ''Mat. Sb.'' , '''86''' :  1  (1971)  pp. 34–75</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Rosenblum,  J. Rovnyak,  "Hardy classes and operator theory" , Oxford Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sarason,  "Operator-theoretic aspects of the Nevanlinna–Pick interpolation problem"  S.C. Power (ed.) , ''Operators and function theory'' , Reidel  (1984)  pp. 279–314</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Kimura,  "Directional interpolation approach in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648037.png" />-optimization"  ''IEEE Trans. Autom. Control'' , '''32'''  (1987)  pp. 1085–1093</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.J.N. Limebeer,  B.D.O. Anderson,  "An interpolation theory approach to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648038.png" /> controller degree bounds"  ''Linear Algebra Appl.'' , '''98'''  (1988)  pp. 347–386</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.A. Ball,  "Nevanlinna–Pick interpolation: Generalizations and applications"  J.B. Conway (ed.) , ''Proc. Asymmetric Algebras and Invariant Subspaces. Conf. Indian Univ.'' , Pitman  (To appear)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Dym,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648039.png" /> contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , Amer. Math. Soc.  (1989)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.W. Helton,  "Operator theory, analytic functions, matrices, and electrical engineering" , Amer. Math. Soc.  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P.L. Duren,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648040.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


Given a class $ \mathfrak H $ of analytic functions in a domain $ G $ of the complex plane (or, in a more general context, of a Riemann surface), to find necessary and sufficient conditions for the solvability in $ \mathfrak H $ of the interpolation problem

$$ \tag{1 } f ( z _ \alpha ) = w _ \alpha , $$

where $ \{ z _ \alpha \} $ is a subset of $ G $, $ \{ w _ \alpha \} $ is some set of complex numbers, and $ \alpha $ usually runs through a countable (sometimes finite, sometimes even uncountable) index set. The classical result of G. Pick [1] and R. Nevanlinna [2] (for finite and countable subsets $ \{ z _ \alpha \} \subset G $, respectively) yields the solution of this problem, for example, in the class $ B _ {1} $ of analytic functions in the unit disc that are bounded by 1 in absolute value. The desired condition here is the non-negativity of the quadratic forms

$$ \sum _ {j, k = 1 } ^ { n } \frac{1 - w _ {j} \overline{ {w _ {k} }}\; }{1 - z _ {j} \overline{ {z _ {k} }}\; } \xi _ {j} \overline{ {\xi _ {k} }}\; ,\ \ n \in \mathbf N ,\ \ \xi _ {j} \in \mathbf C . $$

The first proof of this proposition (see [3], [4]), as well as quite analogous and similar results for various other function classes $ \mathfrak H $, relied on algebraic and functional-theoretic methods. Later proofs, based, for example, on reducing the Nevanlinna–Pick problem to a moment problem or obtained from the point of view of the theory of Hilbert spaces, have made it possible to extend the result to uncountable subsets $ \{ z _ \alpha \} \subset G $ and pointed the way to possible generalizations (see [3][5]).

A natural development of the Nevanlinna–Pick problem, which necessitated an appeal to functional-analytic methods of investigation, was the question of the solvability of the interpolation problem (1) on a class $ W $ of right-hand sides $ \{ w _ \alpha \} $; in this case, as a rule, $ \{ z _ \alpha \} $ is a countable set (a sequence) of points of $ G $, while $ W $ may be one of various spaces of sequences of complex numbers. In connection with the class $ H ^ \infty $ of bounded analytic functions in the unit disc and the space $ l _ \infty $ of bounded sequences, a complete description of the corresponding point sequences $ \{ z _ \alpha \} $( known as universal interpolation sequences) has been obtained (see [6]) in the form of the condition

$$ \tag{2 } \prod _ {\begin{array}{c} j = 1 \\ j \neq k \end{array} } ^ \infty \left | \frac{z _ {j} - z _ {k} }{1 - z _ {j} \overline{ {z _ {k} }}\; } \right | \geq \delta > 0,\ \ k \in \mathbf N . $$

This result played an important role in describing the structure of the maximal ideal space of the algebra $ H ^ \infty $( see [7]) and was at the same time a starting point for extensive research into the Nevanlinna–Pick problem (in the above generalized formulation) for the Hardy classes $ H ^ {q} $ and the spaces $ l _ {p} $( including weight spaces). It turned out that when $ q = \infty $ the solution is independent of $ p $ and is given by condition (2), while when $ q < \infty $ it necessarily varies when $ q $ and $ p $ are changed (see ). Another generalization of the Nevanlinna–Pick problem is connected with the interpolation problem $ \phi _ \alpha ( f ) = w _ \alpha $, where $ \{ \phi _ \alpha \} $ is some system of functionals in a class $ \mathfrak H $. The problem of describing the set $ \{ \{ \phi _ \alpha ( f ) \} : f \in \mathfrak H \} $ may be regarded as a generalization of the well-known coefficient problem for classes of analytic functions.

References

[1] G. Pick, "Ueber die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden" Math. Ann. , 77 (1916) pp. 7–23
[2] R. Nevanlinna, "Ueber beschränkte analytische Funktionen" Ann. Acad. Sci. Fenn. Ser. A , 32 : 7 (1929) pp. 1–15
[3] M.G. Krein, A.A. Nudel'man, "The Markov moment problem and extremal problems" , Amer. Math. Soc. (1977) (Translated from Russian)
[4] J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[5] B. Sz.-Nagy, A. Korányi, "Rélations d'un problem de Nevanlinna et Pick avec la théorie des opérateurs de l'espace Hilbertien" Acta Math. Acad. Sci. Hung. , 7 (1956) pp. 295–303
[6] L. Carleson, "An interpolation problem for bounded analytic functions" Amer. J. Math. , 80 : 4 (1958) pp. 921–930
[7] S.V. Shvedenko, "On -interpolation sequences in the unit disk" Math. USSR Sb. , 46 : 4 (1983) pp. 473–492 Mat. Sb. , 118 : 4 (1982) pp. 470–489

Comments

This classical topic was reactivated and extended to matrix-valued functions in the 1960s and the beginning of the 1970s. Connections with operator theory became essential in the new developments (see, e.g., [a1][a3]). Applications to problems in control theory, which appeared in the 1980s, required a revision of the theory and the development of computational methods, in particular, for rational matrix functions (see [a4], [a5]).

See also $ H ^ \infty $ control theory.

References

[a1] V.M. [V.M. Adamyan] Adamjan, D.Z. Arov, M.G. Krein, "Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Tagaki problem" Math. USSR Sb. , 15 (1971) pp. 31–73 Mat. Sb. , 86 : 1 (1971) pp. 34–75
[a2] M. Rosenblum, J. Rovnyak, "Hardy classes and operator theory" , Oxford Univ. Press (1985)
[a3] D. Sarason, "Operator-theoretic aspects of the Nevanlinna–Pick interpolation problem" S.C. Power (ed.) , Operators and function theory , Reidel (1984) pp. 279–314
[a4] H. Kimura, "Directional interpolation approach in -optimization" IEEE Trans. Autom. Control , 32 (1987) pp. 1085–1093
[a5] D.J.N. Limebeer, B.D.O. Anderson, "An interpolation theory approach to controller degree bounds" Linear Algebra Appl. , 98 (1988) pp. 347–386
[a6] J.A. Ball, "Nevanlinna–Pick interpolation: Generalizations and applications" J.B. Conway (ed.) , Proc. Asymmetric Algebras and Invariant Subspaces. Conf. Indian Univ. , Pitman (To appear)
[a7] H. Dym, " contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , Amer. Math. Soc. (1989)
[a8] J.W. Helton, "Operator theory, analytic functions, matrices, and electrical engineering" , Amer. Math. Soc. (1987)
[a9] P.L. Duren, "Theory of spaces" , Acad. Press (1970)
How to Cite This Entry:
Nevanlinna-Pick problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nevanlinna-Pick_problem&oldid=47962
This article was adapted from an original article by S.V. Shvedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article