Answering a conjecture of M. von Golitschek in the negative, a compact set K is constructed on the plane such that not every continuous function on K can be uniformly approximated by continuous functions of the form g(x)+h( y), and yet K does not contain a closed path of points with consequitive points connected with alternatively horizontal and vertical segments.
1998 Academic Press
Let K be a compact set on the plane. It is a fascinating problem connected with the geometry of K to determine what continuous functions F(x, y) on K can be approximated by tensor-sum functions of the form g(x)+h( y) with continuous g and h (for tensor product spaces in general see [1], for many different applications of this type of approximation see the paper [2] by M. van Golitschek). In particular, when is it true that every continuous F on K can be uniformly approximated by such tensor sums? It is easy to see that if K contains a sequence of distinct points P 1 (x 1 , y 1 ), P 2 (x 1 , y 2 ), P 3 (x 2 , y 2 ), P 4 (x 2 , y 3 ), P 5 (x 3 , y 3 ), ..., P 2k&1 (x k , y k ), P 2k (x k , y 1 ), i.e., for which the line segments P j P j+1 (P 2k+1 =P 1 ) are alternatively vertical and horizontal, then there are functions F that are not approximable. In fact, it is enough to note that for any function F(x, y)= g(x)+h( y) the sum
is zero, so, e.g., if F(P 1 )=1 and F(P j )=0, j=2, ..., 2k, then F cannot be approximated with error less than 1Γ2k by any function F(x, y).