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

AbstractBy expanding a function f∈Lp((-1,1),(1-s)α(1+s)βds) (1≤p<∞) in terms of the basis conjugate to the Jacobi polynomials, the associated generalized translation Tt and the generalized difference T˜t 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