发表机构
University of Bath（巴斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于Helmholtz方程和Neumann边界条件构造Fourier-Bessel小波族，推导归一化常数与闭式傅里叶域表示，旨在探索均匀径向频率分配的应用潜力。
AI 中文摘要
这些笔记发展了受Shaqfa等人[9]的圆盘谐波启发的Fourier-Bessel小波族的数学基础和构造。我们首先介绍Bessel函数和修正Bessel函数的相关性质，并引入构造所需的小波性质。然后，我们将Fourier-Bessel圆盘谐波推导为在单位圆盘上满足Neumann边界条件的Helmholtz方程的解。在此基基础上，我们通过施加高斯空间包络并对第零角阶引入零均值修正来构造一个小波族。我们推导了用于$L^2$应用的相应归一化常数，并讨论了用于频域峰值一致性的$L^1$归一化。最后，我们推导了所得小波的闭式傅里叶域表示。主要动机是近似线性的间距，其收敛于连续径向特征值之间的$\pi$。这一构造并非取代小波族的传统二进组织，而是为探索更均匀的径向频率分配是否可用于需要广泛且平衡频率覆盖的应用奠定基础。
英文摘要
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.
