Un résultat de division obtenu par les méthodes de l'approximation (Q800508)

In this paper, we establish a result of the following type \((\|\cdot \|\) denotes the supremum norm on [-1,1]): Theorem. There exists a constant A such that for any \(g\in C^ p([-1,1])\) satisfying \(g(x)=f(x)| x|^ a\) with f continuous and \(a\in {\mathbb{R}},0<a<p,\) we have either \(\| f\|\leq A\| g\|\) or \(\| f\|\leq A\| g^{(p)}\|^{a/p}\| g\|^{1-a/p}.\) In fact this theorem is proved for g in Lipschitz spaces and satisfying a little more complicated conditions. The proof is done for polynomial functions and extented to other functions by a process of approximation involving Jackson's theorem.