Systems of Dilated Functions: completeness, minimality, basisness

TL;DR

This paper investigates the properties of dilated systems formed by a trigonometric polynomial in various L^p spaces, focusing on their completeness, minimality, and basisness, and discusses known results and open questions.

Contribution

It provides new insights into the completeness, minimality, and basisness of dilated systems in different L^p spaces, highlighting specific conditions and unresolved issues.

Findings

Dilated systems can be complete or minimal depending on the polynomial coefficients.

The basisness of these systems varies with the space and polynomial parameters.

Some cases remain open problems in the theory of dilated systems.

Abstract

We discuss completeness, minimality, and basisness, in $L^2[0, \pi]$ and $L^p[0, \pi]$, $p \neq 2$, of dilated systems $u_n(x) = S(nx)$, $n \in \mathbb{N}$, where $S$ is a trigonometric polynomial $S(x) = \sum_{k = 0}^m a_k \sin(kx), \quad a_0 a_m \neq 0.$ We will present some results and mention a few unsolved questions.