Generalized interpolation in the unit ball (Q5940013)

Let \(B=B^n\) be the unit ball of \(\mathbb C^n\), \(Z=(z_k)_{k \in\mathbb{N}}\) a sequence of \(B\) and \(H^\infty(B)\) the set of all bounded holomorphic functions in \(B\). \(Z\) is called a free interpolating sequence for \(H^\infty (B)\) if for all sequences of values \((w_k)_{k\in \mathbb{N}}\in \ell^\infty\) there exists \(f\in H^\infty (B)\) such that \(f(z_k)=w_k\) for all \(k\in\mathbb N\). Put NEWLINE\[NEWLINE{\mathcal V}^1(Z)= \left\{w= (w_k)_{k\in \mathbb N}\text{ such that }\|w\|_{{ \mathcal V}^1(Z)}: \max\left( \sup_{k\neq j}{|w_k-w_j|\over\rho (z_k,z_j)}, \sup_{k\in \mathbb N}|w_k|\right)< \infty\right\}.NEWLINE\]NEWLINE The sequence \(Z\) is called an interpolating sequence of order 1 for \(H^\infty(B)\) if for all \(w\in {\mathcal V}(Z)\) there exists \(f\in H^\infty(B)\) such that \(f\mid Z=w\). When \(n=2\), the author shows that a sequence \(Z=\{z_k\}\) is an interpolating sequence of order 1 if and only if a union of 3 free interpolating sequences for \(H^\infty (B)\) such that all triples of \(Z\) made of 3 nearby points have to define an angle uniformly bounded below (in an appropriate sense). Moreover he shows that the sequence \(Z\) is a free interpolating sequence if and only if \(Z\) is a multiple interpolating sequence of arbitrary order.