Kolmogorov’s superpositions enable the representation of every real-valued continuous function f defined on the Euclidean n-cube in the form , with continuous functions that compute f, and fixed continuous functions dependent only on n. The functions specify space-filling curves that determine characteristics that are not suitable for efficient computational algorithms. Reversing the process, we specify suitable space-filling curves that enable new functions that give a computational algorithm better adaptable to applications. Detailed numerical constructions are worked out for the case n = 2.
TopIntroduction
This chapter concerns the computation of continuous functions of variables with superpositions of continuous functions of variables. The function
of two variables can offers a simple example of a representation with superpositions of functions of one variable:
where
and
i.e.,
In general, we are interested in representations
for some number
r of summands, in which
are fixed continuous functions that are independent of
.
In their most general form, such superpositions belong to a class of problems conveniently described by means of a commuting diagram with metric spaces X, Y, and T, and a given mapping (Figure 1): The problem is to find continuous mappings and such that f can be replaced with the superposition , where is a fixed embedding of X into T, and when this representation is valid for a sufficiently large class of functions f. We are not going to pursue here this aspect of generality, and instead will adhere to the setting , in which
Figure 1. Superpositions as a commuting diagram
is the
n-dimensional Euclidean unit cube,
,
is the
r-dimensional Euclidean space, and
.