On the approximation by compositions of fixed multivariate functions with univariate functions
The approximation in the uniform norm of a continuous function f(x) = f(x1, . . . , xn) by continuous sums g1(h1(x)) + g2(h2(x)), where the functions h1 and h2 are fixed, is considered. A Chebyshev type criterion for best approximation is established in terms of paths with respect to the functions h1 and h2. 1. Exposition of the problem. It is well known that in many problems of approximation of bivariate functions by sums of univariate functions the concept of a path is central. A path is a finite or infinite ordered set of points in the xy plane such that the line segments joining consecutive points are of positive length and are alternately parallel to the x and y axes. The idea of paths, in this context, was first introduced by Diliberto and Straus [4] and exploited further in a number of works, e.g. [5, 7, 8, 10, 13]. In connection with the problem of interpolation by linear combinations of ridge functions, Braess and Pinkus [1] introduced the notion of a path with respect to distinct directions a and b. This is an ordered set of points (v1, . . . ,v) ⊂ R2 with edges vivi+1 in alternating directions a and b. These objects give a geometric method for deciding if a set of points {x}i=1 has the NI-property (non-interpolation property) (see [1]). Our aim is to bring into consideration more general objects: paths with respect to two continuous functions. We will show how these objects appear in the characterization of extremal elements in the approximation problem considered below. Let Q be a compact set in R. Consider the approximation of a function f ∈ C(Q) by elements of the set Ch1h2 = Ch1h2(Q) = {g ∈ C(Q) : g(x) = g1(h1(x)) + g2(h2(x))}, where the functions hi ∈ C(Q) are prescribed and we vary over functions 2000 Mathematics Subject Classification: 41A30, 41A50, 41A65.