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Register a new userOn the Nevanlinna problem - Characterization of all Schur-Agler class solutions
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Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,ldots, z_nin Omega$ and $w_1,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the $Pick, interpolation, problem$ asks when there is a holomorphic function $f:Omega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinnas result to an arbitrary set $Omega$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.
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Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,z_2,ldots, z_nin Omega$ and $w_1,w_2,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the textit{Pick interpolation problem} asks when there is a holomorphic function $f:Omega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that textit{interpolate} the data. We generalize Nevanlinnas result to a domain $Omega$ in $mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${rho}_{1}^{(r)}(n)=(2rn)!$ and ${rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $int_{0}^{infty}x^{n}W^{(r)}_{1,2}(x)dx = {rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${rho}_{1,2}^{(r)}(n)$, such as the product ${rho}_{1}^{(r)}(n)cdot{rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
In this paper we investigate finitely generated ideals in the Nevanlinna class. We prove analogues to some known results for the algebra of bounded analytic functions $H^{infty}$. We also show that, in contrast to the $H^{infty}$-case, the stable rank of the Nevanlinna class is strictly bigger than 1.
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ partial_t u (x,t) = sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) qquad (x,t) in mathbb{R}^N times, ]- infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.
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