On the expansion of solutions of Laplace-like equations into traces of separable higher dimensional functions

Abstract

AbstractThis paper deals with the equation $$-\varDelta u+\mu u=f$$ - Δ u + μ u = f on high-dimensional spaces $${\mathbb {R}}^m$$ R m where $$\mu $$ μ is a positive constant. If the right-hand side f is a rapidly converging series of separable functions, the solution u can be represented in the same way. These constructions are based on approximations of the function 1/r by sums of exponential functions. The aim of this paper is to prove results of similar kind for more general right-hand sides $$f(x)=F(Tx)$$ f ( x ) = F ( T x ) that are composed of a separable function on a space of a dimension n greater than m and a linear mapping given by a matrix T of full rank. These results are based on the observation that in the high-dimensional case, for $$\omega $$ ω in most of the $${\mathbb {R}}^n$$ R n , the euclidian norm of the vector $$T^t\omega $$ T t ω in the lower dimensional space $${\mathbb {R}}^m$$ R m behaves like the euclidian norm of $$\omega $$ ω .