On the Fourier–Dunkl Coefficients of Generalized Lipschitz Classes on the Interval $$[-1,1]$$

In this paper, we consider \(\mathcal {E}\) the set of all infinitely differentiable functions with compact support included on the interval \(I=[-1,1]\). We use the distributions in \(\mathcal {E}\), as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on I in terms of the behavior of their Fourier–Dunkl coefficients are derived.