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Directional Stockwell transform of distributions

Astrit Ferizi, Katerina Hadzi-Velkova Saneva

TL;DR

The paper develops a directional Stockwell framework that combines directional time–frequency localization with Radon-slice analysis to capture oriented singularities in multi-dimensional signals. It proves an extended Parseval identity, a reconstruction formula, and continuity results for the directional Stockwell transform and its synthesis operator, and places the transform within a robust distributional setting on Lizorkin spaces. By extending the theory to $\mathcal{S}_0'(\mathbb{R}^n)$ and $D_{L^1}'(\mathbb{R}^n)$, it provides a versatile tool for analyzing distributions with directional content in applications such as imaging and signal processing. The work thus broadens the mathematical foundation of directionally sensitive time–frequency analysis and enables stable, duality-based reconstruction in a distributional context.

Abstract

We introduce and study the directional Stockwell transform as a hybrid of the directional short-time Fourier transform and the ridgelet transform. We prove an extended Parseval identity and a reconstruction formula for this transform, as well as results for the continuity of both the directional Stockwell transform and its synthesis transform on the appropriate space of test functions. Additionally, we develop a distributional framework for the directional Stockwell transform on the Lizorkin space of distributions $\mathcal{S}_{0}'(\mathbb{R}^n)$.

Directional Stockwell transform of distributions

TL;DR

The paper develops a directional Stockwell framework that combines directional time–frequency localization with Radon-slice analysis to capture oriented singularities in multi-dimensional signals. It proves an extended Parseval identity, a reconstruction formula, and continuity results for the directional Stockwell transform and its synthesis operator, and places the transform within a robust distributional setting on Lizorkin spaces. By extending the theory to and , it provides a versatile tool for analyzing distributions with directional content in applications such as imaging and signal processing. The work thus broadens the mathematical foundation of directionally sensitive time–frequency analysis and enables stable, duality-based reconstruction in a distributional context.

Abstract

We introduce and study the directional Stockwell transform as a hybrid of the directional short-time Fourier transform and the ridgelet transform. We prove an extended Parseval identity and a reconstruction formula for this transform, as well as results for the continuity of both the directional Stockwell transform and its synthesis transform on the appropriate space of test functions. Additionally, we develop a distributional framework for the directional Stockwell transform on the Lizorkin space of distributions .
Paper Structure (9 sections, 10 theorems, 74 equations)

This paper contains 9 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and spaces
  4. One-dimensional Stockwell transform
  5. The Radon transform
  6. Directional Stockwell transform of functions
  7. Continuity of the directional Stockwell transform on test function spaces
  8. Directional Stockwell transform on $\mathcal{S}_0'(\mathbb{R}^n)$
  9. Directional Stockwell transform on $D_{L^1}'(\mathbb{R}^n)$

Key Result

Proposition 3.2

For $\psi\in \mathcal{S}(\mathbb{R})$ and $f\in L^1(\mathbb{R}^n)$ is true

Theorems & Definitions (22)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 12 more