On the translates of general dyadic systems on $\mathbb{R}$

TL;DR

This paper generalizes the concept of adjacent dyadic systems in harmonic analysis on the real line, providing a classification based on shifts and locations, which helps understand when continuous objects can be decomposed into dyadic counterparts.

Contribution

It introduces a generalized notion of adjacent dyadic systems, characterizes when they occur, and classifies dyadic grids based on shift and location parameters.

Findings

Every dyadic grid is determined by shift and location.

Two dyadic grids form an adjacent system if their parameters satisfy certain conditions.

The classification leads to new insights in harmonic analysis techniques.

Abstract

Many techniques in harmonic analysis use the fact that a continuous object can be written as a sum (or an intersection) of dyadic counterparts, as long as those counterparts belong to an adjacent dyadic system. Here we generalize the notion of adjacent dyadic system and explore when it occurs, leading to some new and perhaps surprising classifications. In particular, we show that every dyadic grid is determined by two parameters, the \emph{shift} and the \emph{location}; moreover two dyadic grids form an adjacent dyadic system if and only if their shifts and locations satisfy readily verifiable conditions.