Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions
In this interesting paper the authors pose some problems that arise from the following classical results: Theorem 1 (R. P. Boas jun., 1935). Suppose \(f\in C^{\infty}(I)\) where I is a real interval. Suppose further that, for some \(r>0\), the sequence \(r^ n(f^{(n)}(x)/n!)\) is bounded, for each \(x\in I\). Then f is analytic on I. Theorem 2 (E. Corominas and F. Sunyer Balaguer, 1954). Suppose \(f\in C^{\infty}(I)\). If, for each \(x\in I\), there is a nonnegative integer \(n=n(x)\) such that \(f^{(n)}(x)=0\), then f is a polynomial. In the paper under review multivariate analogues of both theorems are given as well as their improvements in rather different directions. Reviewer's remark: The authors conjecture the following: Suppose D is a connected open subset of \({\mathbb{R}}^ n\), \(f\in C^{\infty}(D)\) and that, for each \(x\in D\), there is some \(m=m(x)\in {\mathbb{N}}\) such that \(f^{\alpha}(x)=0\) for all \(\alpha \in {\mathbb{N}}^ n\) satisfying \(| \alpha | =n\). Then f is a polynomial. Note that this easily follows from Theorem 2 and from the fact that a function (on \({\mathbb{R}}^ n)\) that is a polynomial with respect to each variable separately must be a polynomial.
scientific article; zbMATH DE number 4120450