Existence of best approximations by exponential sums in several independent variables

AbstractIn this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp(D, where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval