On polyorthogonal functions of the first type

In pre-Hilbert function spaces generated by the measures $\mu_1,\dots,\mu_k$, the process of polyorthogonalization of an arbitrary linearly independent system of functions $\{\varphi_0(x), \varphi_1(x),\dots, \varphi_m(x)\}$ is described, which allows us to introduce the concept of the $n$th polyorthogonal function for an arbitrary multi-index $n$. Necessary and sufficient conditions are found under which this polyorthogonal function is uniquely determined, and its explicit form is described. The main theorem is a multiple analogue of the Gram–Schmidt orthogonalization theorem.