A note on the problem of straight-line interpolation by ridge functions

Abstract

In this paper we discuss the problem of interpolation on straight lines by linear combinations of ridge functions with fixed directions. By using some geometry and/or systems of linear equations, we constructively prove that it is impossible to interpolate arbitrary data on any three or more straight lines by sums of ridge functions with two fixed directions. The general case with more straight lines and more directions is reduced to the problem of existence of certain sets in the union of these lines.

Figures (3)

• Figure 1: In the first picture, the points $A,B,C,D,E,F,G$, in the given order, form a path with respect to the coordinate directions. The second picture illustrates a closed path.
• Figure 2: The six-point closed path $(A,B,C,D,E,F)$ in $l_{1}\cup l_{2}\cup l_{3}$.
• Figure 3: Four-point closed path in $l_{1}\cup l_{2}\cup l_{3}$. The vertex $v(P)$ depends on $P$. There exists $P \in l_{1}\cup l_{2}$ such that $v(P) \in l_{3}$.