Hilbert’s 13th problem asked whether every continuous multivariate function can be written
as superposition of continuous functions of 2 variables. Kolmogorov and Arnold show that
every continuous multivariate function can be represented as superposition of continuous
univariate functions and addition in a universal form and thus solved the problem positively.
In Kolmogorov’s representation, only one univariate function (the outer function)
depends on and all the other univariate functions (inner functions) are independent of the
multivariate function to be represented. This greatly inspired research on representation
and superposition of functions using Kolmogorov’s superposition theorem (KST).
However, the numeric applications and theoretic development of KST is considerably
limited due to the lack of smoothness of the univariate functions in the representation.
Therefore, we investigate the properties of the outer and inner functions in detail. We show
that the outer function for a given multivariate function is not unique, does not preserve the
positivity of the multivariate function and has a largely degraded modulus of continuity.
The structure of the set of inner functions only depends on the number of variables of the
multivariate function. We show that inner functions constructed in Kolmogorov’s representation
for continuous functions of a fixed number of variables can be reused by extension
or projection to represent continuous functions of a different number of variables.
After an investigation of the functions in KST, we combine KST with Fourier transform
and write a formula regarding the change of the outer functions under different inner
functions for a given multivariate function. KST is also applied to estimate the optimal
cost between measures in high dimension by the optimal cost between measures in low
dimension. Furthermore, we apply KST to image encryption and show that the maximal
error can be obtained in the encryption schemes we suggested.