View source for Nevanlinna-Pick problem

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

<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>

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

<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>

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]]]).

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

<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>

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.

====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>



====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]]]).

See also [[H^infinity-control-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066480/n06648036.png" /> control theory]].

====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>