Abstract
Approximative properties of multivariate exponential sums are studied in \(C(D)\) , \(D\) is a nonempty convex compact body in \(\mathbb{R}^d\) . Properties of existence, uniqueness, solarity, and monotone path-connectedness are considered. Several classes of multivariate exponential sums are shown to be proximinal in \(C(D)\) . The set \(\mathbf{E}_n^+\) of multivariate exponential sums with nonnegative coefficients is shown to be a monotone path-connected strict sun in \(C(Q)\) , but not a uniqueness set. We also obtain some negative results on the lack of solarity and monotone path-connectedness for \(E_n\) (the set of extended univariate exponential sums) in \(C[a,b]\) . In particular, we show that \(E_n\) , \(n\ge 2\) , is not monotone path-connected and is not Menger-connected in \(C[a,b]\) . It is also proved that \(E_n\) , \(n> 2\) , is not a sun in \(C[a,b]\) .
