On some properties of finite sums of ridge functions defined on convex subsets of ℝn

Abstract

Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets E of the space Rⁿ. It is shown that under some constraints imposed on the summed functions ϕ_i, in the case when E is open, the continuity of the sum implies the continuity of all ϕ_i. In the case when E is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions ϕ_i in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.