Jacob's ladders and the $\tilde{Z}^2$-transformation of the orthogonal system of trigonometric functions

TL;DR

This paper introduces a new continuum of orthogonal systems based on the weight function Z^2(t), which cannot be derived from existing theories, expanding the understanding of orthogonal systems in analysis.

Contribution

It presents a novel continuum of orthogonal systems relative to Z^2(t), not obtainable through known classical theories.

Findings

Existence of a continuum set of orthogonal systems with weight Z^2(t)

These integrals are outside the scope of Balasubramanian, Heath-Brown, and Ivic theories

New methods for analyzing orthogonal systems with non-standard weights

Abstract

It is shown in this paper that there is a continuum set of orthogonal systems relative to the weight function $\tilde{Z}^2(t)$. The corresponding integrals cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.