On the kernel of the $(\kappa ,a)$ -Generalized fourier transform
For the kernel $B_{\kappa ,a}(x,y)$ of the $(\kappa ,a)$ -generalized Fourier transform $\mathcal {F}_{\kappa ,a}$ , acting in $L^{2}(\mathbb {R}^{d})$ with the weight $|x|^{a-2}v_{\kappa }(x)$ , where $v_{\kappa }$ is the Dunkl weight, we study the important q...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000695/type/journal_article |
| Summary: | For the kernel
$B_{\kappa ,a}(x,y)$
of the
$(\kappa ,a)$
-generalized Fourier transform
$\mathcal {F}_{\kappa ,a}$
, acting in
$L^{2}(\mathbb {R}^{d})$
with the weight
$|x|^{a-2}v_{\kappa }(x)$
, where
$v_{\kappa }$
is the Dunkl weight, we study the important question of when
$\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$
. The positive answer was known for
$d\ge 2$
and
$\frac {2}{a}\in \mathbb {N}$
. We investigate the case
$d=1$
and
$\frac {2}{a}\in \mathbb {N}$
. Moreover, we give sufficient conditions on parameters for
$\|B_{\kappa ,a}\|_{\infty }>1$
to hold with
$d\ge 1$
and any a. |
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| ISSN: | 2050-5094 |