Some constructive properties of functions defined on the sphere

The norm of difference operators for functions defined on the sphere is investigated. A mistake in Rustamov's results is pointed out by a counterexample. And correct result is given. Results of convergence in norm of difference operators acting on the differentiable functions on the sphere are obtained. The highest possible degree tending to zero for the moduli of continuity of fractional order for non-constant valued functions are discussed. The work is supported by NSF of China, Grant No. 19771009.