# Representing systems of dilations and translations in symmetric spaces

## TL;DR

This paper characterizes when dyadic dilations and translations of a function form a representing system in symmetric spaces, using the multiplicator space and frame approach, with sharp conditions identified.

## Contribution

It introduces a criterion involving the multiplicator space for functions to generate representing systems via dilations and translations in symmetric spaces.

## Key findings

- The system forms a representing system if and only if the integral of the function is non-zero and it belongs to the multiplicator space.
- The condition of belonging to the multiplicator space is sharp and necessary.
- For Lorentz spaces, the criterion precisely characterizes functions generating absolutely representing systems.

## Abstract

Let $X$ be an arbitrary separable symmetric space on $[0,1]$. By using a combination of the frame approach and the notion of the multiplicator space $\mathscr{M}(X)$ of $X$ with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function $f\in X$ is a representing system in the space $X$. The main result reads that this holds whenever $\int_0^1 f(t)\,dt\ne 0$ and $f\in \mathscr{M}(X)$. Moreover, the condition $f\in\mathscr{M}(X)$ turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function $f$, $f\ne 0$, from a Lorentz space $\varLambda_{\varphi}$ generates an absolutely representing system of dyadic dilations and translations in $\varLambda_{\varphi}$ if and only if $f\in\mathscr{M}(\varLambda_{\varphi})$.