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Boundary interpolation and rigidity for generalized Nevanlinna functions. (English) Zbl 1210.47040

A Nevanlinna function \(n\) is holomorphic in the upper half plane \({\mathbb C}^+\) with values in \({\mathbb C}^+\cup{\mathbb R}\). It is associated with a positive measure \(\sigma\) on \({\mathbb R}\) (Riesz-Herglotz representation) and a positive kernel \(L(z,w)=[n(z)-n(w)^*]/[z-w^*]\), \(z,w\in{\mathbb C}^+\). A generalized Nevanlinna function allows that the kernel has a number of negative squares. Given the coefficients \(\{\nu_j, j=0,\ldots,2k-1\}\) (moments of \(\sigma\)), the problem is to find all generalized Nevanlinna functions \(n\) such that for \(z\) tending non-tangentially to \(z_1\in{\mathbb R}\) it holds that \(n(z)=\sum_{j=0}^{2k-1}\nu_j (z-z_1)^j +R(z)\) with \(R(z)=o((z-z_1)^{2k-1})\) or \(O((z-z_1)^{2k})\). Note that \(z_1\) is on the boundary of the domain \({\mathbb C}^+\) and hence it is a boundary version of the Nevanlinna-Pick problem where the interpolation values are prescribed in points from \({\mathbb C}^+\). Also, the solutions will be meromorphic with a pole of order \(k\) at \(z_1\), the simplest one being rational of degree \(k\).
The analysis is done in (reproducing kernel) Pontryagin spaces (vector spaces with an indefinite inner product), using properties of the Hankel matrix with entries \(\nu_j\) (this matrix is assumed to be invertible), and the kernel \(L(z,w)\) (which is essentially characterized by the Hankel matrix). The parametrization of the solutions is constructive since it is described as a linear fractional transform of a Nevanlinna function (with some extra condition in \(z_1\)), where the four functions of the transform are the entries of a \(2\times2\) rational \(J\)-unitary matrix (having only poles in \(z_1\)). These entries can be obtained by solving two linear systems whose matrix is the given Hankel matrix.
The rigidity result in the title refers to the conditions where there is only one solution. This paper forms the half-plane analog of [D.Alpay, S.Reich and D.Shoikhet, “Rigidity theorems, boundary interpolation and reproducing kernels for generalized Schur functions”, Comput.Methods Funct.Theory 9, No.2, 347–364 (2009; Zbl 1185.30040)], where Nevanlinna functions were replaced by Schur functions and the half plane by the unit disk.

MSC:

47A57 Linear operator methods in interpolation, moment and extension problems
30E05 Moment problems and interpolation problems in the complex plane
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
30D30 Meromorphic functions of one complex variable (general theory)
15B05 Toeplitz, Cauchy, and related matrices

Citations:

Zbl 1185.30040
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References:

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