Notes on Fourier-Bessel wavelets
Abstract
These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for L2-based applications and discuss L1-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to π between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.
Related
- Frame bounds and bandwidth tiling for time-causal bandpass wavelets
- Vector-Valued Wavelet Bases as Hilbert 𝕄m(ℝ)-Module Bases: A Construction from Scalar Wavelets
- Big Bounce, Big Crunch Cycle: Full Vector--Gravity Coupling, Fixed-Point Instability, and Stable Limit Cycle in a Five-Dimensional Bessel--Klein--Gordon Framework Paper I of a two-part series
- Bessel Distributions and Kloosterman Sums
- Some freezing limits for Bessel functions and Bessel processes with drift of type BN
BibTeX
@misc{venturotti2026notes,
title = {{Notes on Fourier-Bessel wavelets}},
author = {Marcel Venturotti and Georgios Exarchakis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26537},
eprint = {2609.26537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26537},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Notes on Fourier-Bessel wavelets
AU - Marcel Venturotti
AU - Georgios Exarchakis
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26537
UR - https://arxiv.org/abs/2609.26537
AB - These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for L^2-based applications and discuss L^1-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to π between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.
ER -