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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.03554 |
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Table of Contents:
- The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $σ_{x,t}^{k,1}(ξ)$. We will then study the representing measure $σ_{x,t}^{k,1}(ξ)$ and analyze the support of this measure, and derive a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis.