Solutions to a system of equations for $C^m$ functions

Abstract

Fix m\geq 0 , and let A=(A_{ij}(x))_{1 \leq i \leq N, 1\leq j \leq M} be a matrix of semialgebraic functions on \mathbb{R}^n or on a compact subset E \subset \mathbb{R}^n . Given f=(f_1,\ldots,f_N) \in C^\infty(\mathbb{R}^n, \mathbb{R}^N) , we consider the following system of equations: \sum_{j=1}^M A_{ij} (x) F_j (x) = f_i (x) \quad\text{for } i =1,\ldots, N. In this paper, we give algorithms for computing a finite list of linear partial differential operators such that AF=f admits a C^m(\mathbb{R}^n,\mathbb{R}^M) solution F=(F_1,\ldots,F_M) if and only if f=(f_1,\ldots,f_N) is annihilated by the linear partial differential operators.

Locations

• Revista Matemática Iberoamericana - View
• arXiv (Cornell University) - View - PDF