A Bernstein type inequality for the Askey-Wilson operator.

We establish a Riesz-type interpolation formula on the interval [−1,1] for the Askey–Wilson operator. As consequences, sharp Bernstein inequality and Markov inequality are obtained when differentiation is replaced by the Askey–Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein type inequality in L2-space. We conclude our paper with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail.