Quotients of smooth functions
The authors prove, among other things, the following theorems. Theorem 1. Let f,g: \({\mathbb{R}}\to {\mathbb{C}}\) be two functions, m a natural integer, and let c, \(\alpha\) be two positive constants such that (a) \(g,fg,f^ m\in C^{\infty}({\mathbb{R}},{\mathbb{C}})\); (b) \(| f(x)| \leq c| g(x)|^{\alpha}\) for every real x. Then \(f\in C^{\infty}({\mathbb{R}},{\mathbb{C}}).\) Theorem 2. If \(m_ 1,m_ 2,...,m_ n\) and \(s_ 1,s_ 2,...,s_ n\) are natural numbers such that \(g.c.d.(m_ j,2s_ j)=1\), \(j=1,...,n\), and if \(f_ 1,...,f_ n: {\mathbb{R}}\to {\mathbb{R}}\) are such that \[ f_ 1^{m_ 1},...,f_ n^{m_ n},\sum^{n}_{1}f_ j^{2s_ j}\in C^{\infty}, \] then \(f_ 1,...,f_ n\in C^{\infty}.\) Theorem 3. Let f,g: \({\mathbb{R}}\to {\mathbb{C}}\) be two functions and \(\alpha\) a positive constant; suppose that (a) \(g,fg\in C^{\infty}\); (b) \(| f(x)| \leq | g(x)|^{\alpha}\) for all real x. Then \(f\in C^{[2\alpha]}\), if f is real, and \(f\in C^{[\alpha]}\), if f is complex. The result is best possible: there are real functions f, g satisfying (a) and (b), with \(f\not\in C^{[2\alpha]+1}\); there are complex functions f, g satisfying (a) and (b), with \(f\not\in C^{[\alpha]+1}\).
scientific article; zbMATH DE number 4205428