Wentao Teng, Takayuki Okuda,
| Date: | Mar 21 (Fri), 2025, 17:00-17:40 |
| Speaker: | Wentao Teng (The University of Tokyo) |
| Title: | A positive product formula of integral kernels of $k$-Hankel transforms |
| Abstract: [ pdf ] | Let $R$ be a root system in $\mathbb R^N$ and $G$ be the finite subgroup generated by the reflections associated to the root system. We establish a positive radial product formula for the integral kernels $B_{k,1}(x,y)$ of $(k,1)$-generalized Fourier transforms (or the $k$-Hankel transforms) $F_{k,1}$ \[ B_{k,1}(x,z)j_{2\left\langle k\right\rangle+N-2}\left(2\sqrt{t\left|z\right|}\right)=\int_{\mathbb R^N} B_{k,1}(\xi,z)\,d\sigma_{x,t}^{k,1}(\xi), \] where $j_{\lambda}$ is the normalized Bessel function, and $\sigma_{x,t}^{k,1}(\xi)$ is the unique probability measure. Such a product formula is equivalent to the following representation of the generalized spherical mean operator $f\mapsto M_f,\;f\in C_b(\mathbb{R}^N)$ in $(k,1)$-generalized Fourier analysis \begin{align*} M_f(x,t)=\int_{\mathbb{R}^N}f\,d\sigma_{x,t}^{k,1},\;(x,t) \in\mathbb{R}^N\times{\mathbb{R}}_+. \end{align*} We will then analyze the representing measure $\sigma_{x,t}^{k,1}(\xi)$ and show that the support of the measure is contained in \[ \left\{\xi\in\mathbb{R}^N:\sqrt{\vert\xi\vert}\geq\vert\sqrt{\vert x\vert}-\sqrt t\vert\right\}\cap\left(\bigcup_{g\in G}\{\xi\in\mathbb{R}^N:d(\xi,gx)\leq\sqrt t\}\right), \] where $d\left(x,y\right)=\sqrt{\left|x\right|+\left|y\right|-\sqrt{2\left(\left|x\right|\left|y\right|+\left\langle x,y\right\rangle\right)}}$. Based on the support of the representing measure $\sigma_{x,t}^{k,1}$, we will get a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis. Moreover, for $N\geq 2$, if we assume that $F_{k,1}\left(\mathcal S(\mathbb{R}^N)\right)$ consists of rapidly decreasing functions at infinity, then we get two different results on $\text{supp}\sigma_{x,t}^{k,1}$, which indirectly denies such assumption. |
| Joint with Tuesday Seminar on Topology | |
| Date: | Apr 22 (Tue), 2025, 17:30-18:30 |
| Speaker: | Takayuki Okuda (奥田 隆幸) (Hiroshima University) |
| Title: | Coarse coding theory and discontinuous groups on homogeneous spaces |
| Abstract: [ pdf ] | Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map \[ R : M \times M \to \mathcal{I}. \] For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\ mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying \[ R(C \times C) \cap \mathcal{A} = \emptyset. \] This definition encompasses various types of codes, including error- correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\ mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$. |
© Toshiyuki Kobayashi