Remarks on Dunkl translations of non-radial kernels

2022/11/04 by Jacek Dziubański, Agnieszka Hejna, Dziubański, Jacek +1 · 1 citation
Mathematics · Medicine · #33C52 #35K08 #39A70 #42B20 #42B25 #44A20 #47B38 #Advanced Algebra and Geometry #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications

paper · pdf · doi:10.48550/arxiv.2211.02518

openalex publication_date 2022/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On \mathbb R^N equipped with a root system R and a multiplicity function k>0, we study the generalized (Dunkl) translations τ_{\mathbf x}g(-\mathbf y) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for τ_{\mathbf x}g(-\mathbf y) by means the Euclidean distance ‖\mathbf x-\mathbf y‖ and the distance d(\mathbf x,\mathbf y)=min_{σ∈ G} ‖ \mathbf x-σ(\mathbf y)‖, where G is the reflection group associated with R. Moreover, we prove that τ does not preserve positivity, that is, there is a non-negative Schwartz class function φ, such that τ_{\mathbf x}φ(-\mathbf y)<0 for some points \mathbf x,\mathbf y∈\mathbb R^N.

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