Jacob's ladders and vector operator producing new generations of $L_2$-orthogonal systems connected with the Riemann's $\zeta(\frac 12+it)$ function

Abstract

In this paper we introduce a generating vector-operator acting on the class of functions $L_2([a,a+2l])$. This operator produces (for arbitrarily fixed $[a,a+2l]$) infinite number of new generation $L_2$-systems. Every element of the mentioned systems depends on Riemann's zeta-function and on Jacob's ladder.