Optimal Arguments in the Jackson–Stechkin Inequality in $L_2(\mathbb{R}^d)$ with Dunkl Weight

The paper is devoted to the determination of the optimal arguments in the sharp Jackson–Stechkin inequality with modulus of continuity of order $r$ in the space $L_2(\mathbb{R}^d)$ with Dunkl weight defined by the root system $R$ and a nonnegative function of multiplicity $k$. If
$$ \lambda_k=\frac d2-1+\sum_{\alpha\in R_+}k(\alpha)=\frac12, $$
where $R_+$ is the positive subsystem of the root system, then the optimal arguments for all $r$ coincide. If $\lambda_k\ne 1/2$, then the optimal argument for the modulus of continuity of second order is greater than for the first order. Such patterns are related to the arithmetic properties of zeros of Bessel functions.