Existence of good and best approximations on unbounded domains by exponential sums in several independent variables

AbstractIn this paper we consider the problem of using exponential sums to approximate a given complex-valued function f defined on the possibly unbounded domain D in Rm. We establish the existence of a best approximation from the set of exponential sums having order at most n and formulate a Weierstrass-type density theorem. In so doing we extend previously known results which apply only in the special cases where D is bounded or where m = 1