We consider two types of geometric graphs on point sets on the plane based on a plane set C: one obtained by translates of C, another by positively scaled translates (homothets) of C. For compact and convex C, graphs defined by scaled translates of C, i.e., Delaunay graphs based on C, are known to be plane graphs. We show that as long as C is convex, both types of graphs are plane graphs.

We study approximation by arbitrary linear combinations of $n$ translates a single function periodic functions. construct some methods this for univariate functions in the class induced convolution with function, and prove upper bounds $L^p$-approximation convergence rate these methods, when $n \to \infty$, $1 \leq p \infty$. also generalize results to classes multivariate defined tensor produc...