# A uniformly bounded complete Euclidean system

## TL;DR

The paper constructs a uniformly bounded complete orthonormal system on [0,1] that exhibits convergence for square-summable coefficients and divergence otherwise, showing Menshov's theorem cannot be extended to such systems.

## Contribution

It introduces a new uniformly bounded complete orthonormal system demonstrating convergence and divergence properties that challenge extensions of Menshov's theorem.

## Key findings

- System converges a.e. for l^2 coefficients
- System diverges a.e. for non-l^2 coefficients
- Limits extension of Menshov's theorem to bounded systems

## Abstract

A uniformly bounded complete orthonormal system of functions $\Theta =\{ \theta_n\}_{n=1}^{\infty},$ $ \|\theta_n\|_{L^\infty_{[0,1]} } \leq M $ is constructed such that $\sum_{n=1}^{\infty} a_{n}\theta_{n}$ converges almost everywhere on $[0,1]$ if $\{ a_n\}_{n=1}^{\infty} \in \, l^2$ and $\sum_{n=1}^{\infty} a_{n}\theta_{n}$ diverges a. e. for any $\{ a_n\}_{n=1}^{\infty} \not\in \, l^2$. Thus Menshov's theorem on the representation of measurable, almost everywhere finite, functions by almost everywhere convergent trigonometric series cannot be extended to the class of uniformly bounded complete orthonormal systems.