Characterizations of smoothness of functions in terms of the basis conjugate to Jacobi polynomials

By expanding a function fL^p((-1,1),(1-s)^α(1+s)^βds) (1≤p<∞) in terms of the basis conjugate to the Jacobi polynomials, the associated generalized translation and the generalized difference are introduced to define the smoothness of f, respectively, The related Lipschitz classes are characterized by the properties of the Poisson integral near to the boundary.