On reflection principles supported on a finite set

We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation ∂ 2 u / ∂ x ∂ y = 0 , there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation ∂ 2 u / ∂ x ∂ y + λ 2 u = 0 , we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.