# On the orthogonal democratic systems in the $L^p$ spaces

## TL;DR

This paper constructs specific orthonormal systems in $L^p$ spaces that exhibit bidemocratic properties depending on a parameter, revealing nuanced behavior of these systems in relation to democratic and bidemocratic criteria.

## Contribution

It introduces a family of orthonormal systems with parameter-dependent bidemocratic properties for $L^p$ and $L^{p'}$ spaces, expanding understanding of democratic systems in functional analysis.

## Key findings

- The systems are bidemocratic for $L^p$ and $L^{p'}$ when $l$ is within a certain range.
- The systems are not democratic or bidemocratic outside that range.
- The properties depend critically on the parameter $l$ and the space $L^p$ considered.

## Abstract

The concept of bidemocratic pair for a Banach space was introduced in \cite{KS:18}. We construct a family of orthonormal systems $\mathfrak{F}_{l},$ $l\in (0,\infty)$ of functions defined on $[-1,1]$ such that the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for $L^{p'}[-1,1]$ if $l\in (0, \frac{p}{2(p-2)}]$, where $p>2$ and $p'= \frac{p}{p-1}$. The system $\mathfrak{F}_{l}$ is not democratic for $L^{p'}[-1,1]$ when $l\in (\frac{p}{2(p-2)}, \frac{p}{p-2}). $ When $l> \frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p'}[-1,1]$.