Tame extension operators for Hermite's interpolation problem on algebraic varieties (Q1905825)

Let \(V\) be an algebraic variety in complex \(n\)-space \(\mathbb{C}^n\), \(H(\mathbb{C}^n)\) denote the space of entire functions on \(\mathbb{C}^n\), and \(H(V)\) denote the space of holomorphic functions on \(V\). Given a linear first-order differential operator \(L\) in \(\mathbb{C}^n\) with polynomial coefficients, a positive integer \(k\), and \(F\in H(\mathbb{C}^n)\), the authors define the operator \(T_k\) from \(H(\mathbb{C}^n)\) into \(H(V)^{k+ 1}\) by \(T_k F= \{L^j F|V: 0\leq j\leq k\}\). The range \(T^k(H(\mathbb{C}^n))= X_k\) is called the set of admissible data. The authors extend some of their earlier results [Bull. Pol. Acad. Sci., Math 37, No. 7-12, 673-677 (1989; Zbl 0766.32015)] to study the Hermite interpolation problem of finding a right inverse \(S_k\) for \(T_k\). They give a recursive characterization of \(X_k\) and show that the extension operator \(S_k\) satisfies some radial bounds on the growth of its norm. From this they deduce that if the growth of the data in \(X_k\) has exponential bounds, then the growth of the image under \(S_k\) has more or less the same exponential bound.