Interpolation on countably many algebraic subsets for weighted entire functions
Let \(X\) be an analytic subset of \(\mathbb{C}^n\). Let \(\mathcal{O}(X)\) be the space of analytic functions on \(X\) (if \(X=\mathbb{C}^n\) then \(\mathcal{O}(\mathbb{C}^n)\) is the ring of all entire functions on \(\mathbb{C}^n\)) and let \(p\) be a weight function on \(\mathbb{C}^n\). Set \(A_p(X)=\{f\in\mathcal{O}(X)\): There exist constants \(A,B>0\) such that \(|f(z)|\leq A\exp(Bp(z))\) for all \(z\in X\}\). An analytic subset \(X\) of \(\mathbb{C}^n\) is said to be interpolating for \(A_p(\mathbb{C}^n)\) if the restriction map \(R_X\) : \(A_p(\mathbb{C}^n)\to A_p(X)\) defined by \(R_X(f)=f|_X\) is surjective. The main result of this paper is as follows: Suppose that \(m\leq n\). Let \(X=\{\zeta_\nu\}_{\nu\in\mathbb{N}}\) be a discrete variety in \(\mathbb{C}^m\) and let \(F=(F_1,\dots, F_m)\in\mathbb{C}[z_1,\dots, z_n]^m\). Put \(d=\max_{j=1,\dots, m}\)deg \(F_j\). For \(a>0\), we assume that (1) \(X\) is interpolating for \(A_{|\cdot|^a}(\mathbb{C}^m)\); (2) there exist constants \(\varepsilon, C>0\) and a finite subset \(E\) of \(\mathbb{N}\) such that \[ \sum\limits_{k=1}^{\binom nm}|\Delta_k^F(z)|\geq\varepsilon\exp(-C|z|^{ad}) \] for all \(z\in F^{-1}(\zeta_\nu)\), \(\nu\in\mathbb{N}\setminus E\). Here the sum is taken over all \(m\times m\) minors \(\Delta_k^F\) of the Jacobian matrix \(JF\). Then \(F^{-1}(X)\) is interpolating for \(A_{|\cdot|^b}(\mathbb{C}^n)\) for every \(b\geq ad\).
scientific article; zbMATH DE number 1747459