Bernstein and Jackson Inequalities for the Quaternionic Dunkl Transform

Abstract

We study approximation theory for the quaternionic Dunkl transform (QDT) on ℝ². We establish Bernstein-type inequalities and Jackson direct theorems, relating the best approximation Q_ν(f)₂,Q to moduli of smoothness. We also prove inverse theorems showing that summability of Qₙ(f)₂,Q implies Sobolev regularity f ∈ Wₖᵐ(ℝ²,mathbbH). These results extend classical Jackson-Bernstein theory to the quaternionic Dunkl setting and provide a bridge between spectral approximation and Dunkl-Sobolev smoothness.