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Interpolation with gradient in the ball and polydisk. (English. Russian original) Zbl 0976.32009

Math. Notes 66, No. 3, 328-356 (1999); translation from Mat. Zametki 66, No. 3, 407-416 (1999).
Let \(\mathcal{D}^n\) be any of domains \(\mathbb{B}^n\), \(\mathbb{D}^n\), where \(\mathbb{B}^n\) is the unit ball in \(\mathbb{C}^n\) and \(\mathbb{D}^n\) is the unit polydisk in \(\mathbb{C}^n\). Let \(A=\{a_k\in\mathcal{D}^n\}_{k=1}^\infty\) be a fixed sequence of points in \(\mathcal{D}^n\). Denote by \(r(a,m)=(1-||a||^2+|\langle a,\bar{m}\rangle|^2)^{1/2}\). A sequence of complex numbers \(\{f_k\}_{k=1}^\infty\) and vectors \(\{b_k\in\mathbb{C}^n\}_{k=1}^\infty\) are called \(A\)-admissible for the algebra \(H^\infty(\mathcal{D}^n)\) of bounded holomorphic functions in \(\mathcal{D}^n\) if there exist constants \(M_1\), \(M_2>0\) such that
1) \(|f_k|\leq M_1\) for any \(k\in\mathbb{N}\);
2a) in the case of the polydisk: \(\bigl|b_k^s\bigr|\leq M_2/(1-|a_k^s|^2)\) for any \(1\leq s\leq n\), \(k\in\mathbb{N}\);
2b) in the case of the unit ball: \[ |\langle b_k,m\rangle|=\left|\sum\limits_{s=1}^nb_k^sm^s\right|\leq M_2\frac{r(a_k,m)}{ r(a_k,m)^2-|\langle a_k,\bar{m}\rangle|^2}, \] for any unit vector \(m\in\mathbb{C}^n\), \(|m|=1\), and any \(k\in\mathbb{N}\).
A sequence \(A\) of points in \(\mathcal{D}^n\) is called an interpolation sequence for \(H^\infty(\mathcal{D}^n)\) for the problem of interpolation with gradient if for any \(A\)- admissible sequence of complex numbers \(\{f_k\}_{k=1}^\infty\) and vectors \(\{b_k\in\mathbb{C}^n\}_{k=1}^\infty\) there exists a holomorphic function \(F\in H^\infty(\mathcal{D}^n)\) such that \(F(a_k)=f_k\) and \(\partial_k F(a_k)=b_k^s\), \(1\leq s\leq n\).
The author obtains sufficient conditions for the solvability of this problem. In particular, he proves
Theorem. A sequence \(A\) of points in the unit polydisk \(\mathbb{D}^n\) is interpolating for the interpolation problem with gradient in the algebra \(H^\infty(\mathbb{D}^n)\) if there exists a constant \(\delta>0\) such that the inequality \[ \prod\limits_{j:j\neq k}\rho^s(a_j,a_k)\geq\delta \] is true for any \(1\leq s\leq n\) and \(k\in\mathbb{N}\). Here \(\rho\) is hyperbolic metric in \(\mathbb{C}^n\).
Reviewer: K.Malyutin (Sumy)

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
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References:

[1] L. Carleson, ”An interpolation problem for bounded analytic functions,”Amer. J. Math.,8, 921–930 (1958). · Zbl 0085.06504 · doi:10.2307/2372840
[2] B. Berndtsson, ”Interpolating sequences forH in the ball,”Indag. Math.,88, 1–10 (1985). · Zbl 0588.32006
[3] B. Berndtsson, ”Interpolating sequences in the polydisk,”Trans. Amer. Math. Soc.,302, 161–169 (1987). · Zbl 0638.42021 · doi:10.1090/S0002-9947-1987-0887503-9
[4] N. Th. Varopoulos, ”Sur une probléme d’interpolation,”C. R. Acad. Sci. Paris. Sér. A,274, 1539–1542 (1972). · Zbl 0236.41001
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