Our aim in this talk is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz condition in the space $L_p, 1 < p \leq 2$ on the sphere. We also prove analogues of Jackson’s direct theorem for the moduli of smoothness of all orders constructed on the basis of spherical shift. Finally, we prove equivalence between moduli of smoothness and $K$-functional for the couple $(L^2 (\sigma^{m-1} ), W^r_2 (\sigma^{m-1} ))$.

This is joint work with S. El Ouadih, O. Tyr and F. Saadi.