On cross products of orthonormal systems of functions

The continual analog of an orthonormal system of functions is an orthonormal kernel. In this article the concept of cross product of orthonormal systems of functions is introduced, and it is shown that the cross product of any two orthonormal systems which are complete in $L_2$ is a complete orthonormal kernel with respect to Lebesgue measure on half-axes. The properties of the cross product of two orthonormal systems which are complete in $L_2$, each of which is uniformly bounded, are studied, as are the properties of the cross product of a Haar system on an orthonormal system of functions, complete in $L_2$, which are uniformly bounded.