Result: Interpolating varieties for weighted spaces of entire functions in $\mathbf{C}^n$: Interpolating varieties for weighted spaces of entire functions in \(\mathbb{C}^ n\)

Let \(A(\mathbb{C}^ n)\) be the set of all entire functions on \(\mathbb{C}^ n\), let \(p(\xi)\), \(\xi \in \mathbb{C}^ n\), be a plurisubharmonic weight function on \(\mathbb{C}^ n\). A discrete variety \(V \equiv \{\zeta_ k\}\) is an interpolating variety for the set \(A_ p (\mathbb{C}^ n) \equiv \{f \in A (\mathbb{C}^ n) : | f(\xi) | \leq A \exp (Bp(\xi))\}\), for some \(A,B > 0\) if the restriction map \(p\) is onto from \(A_ p (\mathbb{C}^ n)\) to \(A_ p(V) \equiv \{a \equiv \{a_ k\}_{k \in \mathbb{N}} : \exists A,B > 0\), \(| a_ k | \leq A \exp (Bp (\zeta_ k)),\;\forall k \in \mathbb{N}\}\). The main result of this paper is the following Theorem 2.5. The variety \(V\) is an interpolating variety for \(A_ p (\mathbb{C}^ n)\) if and only if there exist \(m(\geq n)\) functions \(f_ 1, f_ 2, \dots, f_ m \in A_ p (\mathbb{C}^ n)\) such that \(V \subseteq \{z : f_ 1(z) \equiv \cdots \equiv f_ m \equiv 0\}\) and for some \(\varepsilon\), \(C > 0\) \[ \sum^ m_{j=1} | D_ u f_ j(\zeta_ k) | \geq \varepsilon (-Cp (\zeta_ k)),\;\forall k \in \mathbb{N}, \] where \(D_ u f(\xi)\) is the directional derivative of \(f\) along the direction \(u\) from unit sphere \(S^{2n-1}\). As a corollary of this theorem the authors obtain another criterium in terms of the Jacobian matrix of \(f_ 1, \dots, f_ m\) and show that if \(V\) is an interpolating variety for \(A_ p (\mathbb{C}^ n)\) then \(V\) is an interpolating variety for \(A_ q (\mathbb{C}^ n)\) if \(q \geq p\).