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Haar wavelet characterization of dyadic Lipschitz regularity

Hugo Aimar, Carlos Exequiel Arias, Ivana Gómez

TL;DR

This work characterizes dyadic Lipschitz regularity on $ℝ^+$ in terms of Haar coefficients. It proves a necessary and sufficient condition: a function $f$ belongs to $Lip_δ(α)$ if and only if its Haar coefficients satisfy $|⟨f,h^j_k⟩| ≤ C 2^{-(α+1/2)j}$ for all dyadic scales $j$ and positions $k$. The proof links the coefficients to dyadic mean oscillations via $|⟨f,h_I⟩|$ and builds a dyadic-chain decomposition of $f(x)-f(y)$ to establish the reverse implication, yielding a concrete constant $C_α$. The results provide a discrete, dyadic-harmonic-analysis criterion for ultrametric Lipschitz regularity on $ℝ^+$ with potential applications to irregular signal analysis and function regularity in ultrametric settings.

Abstract

We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $α$ regularity of $f$ with respect to the ultrametric $δ(x,y)=\inf \{|I|: x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $α$ is positive. Precisely, $f\in \textrm{Lip}_δ(α)$ if and only if $\left\vert\left<f,h^j_k\right>\right\vert\leq C 2^{-(α+ \tfrac{1}{2})j}$, for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$.

Haar wavelet characterization of dyadic Lipschitz regularity

TL;DR

This work characterizes dyadic Lipschitz regularity on in terms of Haar coefficients. It proves a necessary and sufficient condition: a function belongs to if and only if its Haar coefficients satisfy for all dyadic scales and positions . The proof links the coefficients to dyadic mean oscillations via and builds a dyadic-chain decomposition of to establish the reverse implication, yielding a concrete constant . The results provide a discrete, dyadic-harmonic-analysis criterion for ultrametric Lipschitz regularity on with potential applications to irregular signal analysis and function regularity in ultrametric settings.

Abstract

We obtain a necessary and sufficient condition on the Haar coefficients of a real function defined on for the Lipschitz regularity of with respect to the ultrametric , where is the family of all dyadic intervals in and is positive. Precisely, if and only if , for some constant , every and every Here, as usual and .
Paper Structure (3 sections, 4 theorems, 21 equations)

This paper contains 3 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Dyadic distance in $\mathbb{R}^+$ and the Haar system
  3. Proof of Theorem \ref{['thm:mainresult']}

Key Result

Theorem 1.1

Let $f$ be a real valued function in $L^1_{\emph{loc}}(\mathbb{R}^+)$. Let $h^j_k(x)=2^{j/2} h(2^jx-k)$ where $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$, $j\in\mathbb{Z}$, $k=0, 1, 2, \ldots$, and $\left<f,h^j_k\right>=\int_{\mathbb{R}^+}f(x) h^j_k(x) dx$. Let $\alpha$ be any positive is equivalent to the essential boundedness of the quotients where $\delta(x,y)=\inf\{\left\vert I\

Theorems & Definitions (7)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof