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Ridgelet Transforms of Functions in Banach lattices

Mitsuo Izuki, Takahiro Noi, Yoshihiro Sawano, Hirokazu Tanaka

Abstract

We establish a reproducing formula for the ridgelet transform on $\mathbb{R}^n$ in the framework of Banach lattices introduced in a recent paper by Nieraeth. Our approach is based on the $k$-plane Radon transform and a wavelet-type reconstruction operator acting on functions defined on the Grassmannian of $k$-dimensional affine planes. Under mild structural assumptions on the underlying Banach lattice, we prove that the ridgelet reconstruction converges both in the lattice norm and almost everywhere. The admissibility conditions on the wavelet function are formulated in terms of the Riemann--Liouville fractional integral. As a consequence, we obtain explicit inversion formulas for functions in a Banach lattice $X$ which is contained in $L^1({\mathbb R}^n)+L^p(\mathbb{R}^n)$ with some constant $1 \le p < \frac{n}{k}$, together with precise expressions for the reconstruction constant. These results provide a unified framework for ridgelet-type reproducing formulas in a broad class of function spaces beyond the classical $L^p$ setting.

Ridgelet Transforms of Functions in Banach lattices

Abstract

We establish a reproducing formula for the ridgelet transform on in the framework of Banach lattices introduced in a recent paper by Nieraeth. Our approach is based on the -plane Radon transform and a wavelet-type reconstruction operator acting on functions defined on the Grassmannian of -dimensional affine planes. Under mild structural assumptions on the underlying Banach lattice, we prove that the ridgelet reconstruction converges both in the lattice norm and almost everywhere. The admissibility conditions on the wavelet function are formulated in terms of the Riemann--Liouville fractional integral. As a consequence, we obtain explicit inversion formulas for functions in a Banach lattice which is contained in with some constant , together with precise expressions for the reconstruction constant. These results provide a unified framework for ridgelet-type reproducing formulas in a broad class of function spaces beyond the classical setting.
Paper Structure (16 sections, 13 theorems, 136 equations)

This paper contains 16 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Banach lattices
  4. The Linear Grassmannian $G_{n,k}$
  5. The Affine Grassmannian $\mathcal{G}_{n,k}$
  6. The $k$-plane Radon and ridgelet transforms
  7. An observation on an averaging lemma
  8. Proof
  9. Proof of Theorem \ref{['thm:wavelet-reconstruction']}
  10. Proof of Theorem \ref{['thm:wavelet-reconstruction1']}
  11. Proof of Theorem \ref{['Theorem 3.5']}
  12. Examples of admissible and non-admissible function spaces
  13. Lorentz spaces
  14. Lebesgue spaces with variable exponents
  15. Morrey spaces and limitations of the reconstruction theorems
  16. ...and 1 more sections

Key Result

Theorem 1.5

Let $X \in {\mathcal{A}}$ be a Banach lattice satisfying the following conditions: Assume that $\psi$ is a measurable function on $\mathbb{R}_+$, and that there exists an admissible function $w$ such that Assume that $\psi$ admits an integrable decreasing majorant. Define the smoothened dual $k$-plane transform $W_t^*$ by eq:W-star-def. Then the inversion formula holds, where The convergence h

Theorems & Definitions (29)

  • Definition 1.1: $k$-plane Radon transform
  • Definition 1.2
  • Example 1.3
  • Definition 1.4: Muckenhoupt condition
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Corollary 1.8
  • Definition 1.9: $k$-plane ridgelet-type transforms
  • Theorem 1.10
  • ...and 19 more