Remarks on Dunkl translations of non-radial kernels

TL;DR

This paper investigates the behavior of Dunkl translations of non-radial kernels in Euclidean space with reflection group symmetries, providing bounds and showing that these translations do not always preserve positivity.

Contribution

It establishes bounds for Dunkl translations of non-radial kernels and demonstrates that positivity is not preserved under these transformations.

Findings

Bounds for Dunkl translations in terms of Euclidean and group-invariant distances

Dunkl translations can produce negative values from non-negative functions

Non-positivity preservation is demonstrated with explicit examples

Abstract

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