# Nevanlinna-Pick problem

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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

See also $H ^ \infty$ control theory.