A correlation integral equation is an equation of the following form (see the article). The purpose of this paper is to study the system of fundamental functions of the equation. We will carry out the reasoning for the case of symmetric correlation (see the article). However, this restriction can be easily removed and non-symmetric correlation can be considered if we pass to the system of integral equations using the Hilbert-Schmidt theory. Let us prove several theorems (see the article).

Let’s consider a correlation integral equation of the form (see the article). All conditions and remarks concerning this equation are fully preserved. To equations of this kind, as is known, the problem of transforming a curvilinear correlation relation between two continuously distributed random variables into a rectilinear correlation relation between the new variables is reduced. It turns out that the problem of finding the set of all pairs of functions (in the case of symmetric correlation) between which the correlation is rectilinear is equivalent to finding the fundamental functions of the equation.