On restricting Cauchy–Pexider functional equations to submanifolds

Abstract

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in $${\mathbb{R}^d}$$ has only solutions which extend uniquely to exponential affine functions $${\mathbb{R}^d \to \mathbb{C}}$$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations $${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$$ for $${x_0, \ldots,x_d}$$ lying on a curve and $${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$$ for x 1, x 2, x 3 lying on a hypersurface are also considered.