Approximative Properties of Multivariate Exponential Sums

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 $$C[a,b]$$ .