Ridgelet Transforms of Functions in Banach lattices

TL;DR

This paper develops a ridgelet transform framework for functions in Banach lattices, providing explicit inversion formulas and convergence results, extending classical ridgelet analysis beyond standard $L^p$ spaces.

Contribution

It introduces a unified ridgelet reconstruction approach in Banach lattices, with convergence in norm and almost everywhere, and explicit inversion formulas under mild assumptions.

Findings

Reproducing formula for ridgelet transform in Banach lattices established.

Convergence of ridgelet reconstruction proven in lattice norm and almost everywhere.

Explicit inversion formulas derived for functions in certain Banach lattices.

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.