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

In this paper, we consider E the set of all infinitely differentiable func-tions with compact support included on the interval I=[-1,1]. We use the dis-tributions in 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 theDunkl operator are verified. By means of generalized Dunkl-Lipschitz condi-tions on Dunkl-Sobolev spaces, a result of Younis on the torus, which is ananalog of Titchmarsh's theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini-Lipschitz classes onIin termsof the behavior of their Fourier-Dunkl coefficients are derived.