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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.03554 |
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| _version_ | 1866917518397931520 |
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| author | Teng, Wentao |
| author_facet | Teng, Wentao |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A positive product formula of integral kernels of $k$-Hankel transforms Teng, Wentao Classical Analysis and ODEs 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. |
| title | A positive product formula of integral kernels of $k$-Hankel transforms |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.03554 |