Jacobi Series And Smoothness Of Functions

The purpose of this thesis is to characterize smoothness of functions in terms of the asymptotic behaviour of the Poisson integrals of Jacobi series of functions.The function theory related to Jacobi series is an important field in mathematics, and there have been some achievements in study of associated problems. In comparison with the classical Fourier analysis, there are still some deep problems to be considered and studied in the theory of Jacobi series, and because of the complexity of Jacobi polynomials, there would be many difficulties in study of these problems. On the other hand, due to the diversity of parameters, there would be some new type results in this theory.In classical Fourier analysis, there are some results about the characterizations of smoothness of functions in terms of the asymptotic behaviour of the Poisson integrals of functions. called the theory of Hardy-Littlewood and Zygmund. This thesis is to study the theory of Hardy-Littlewood and Zygmund associated with Jacobi series, and to generalize some conclusions about classical Poisson integrals and smoothness of functions to the case of Jacobi series. and in the new case, the smoothness of functions is presented by the generalized translation T_t2 and the generalized difference (T|～)_t2.In the second section, some estimates of Jacobi-Poisson integrals are obtained, and the asymptotic dependence of norms of conjugate Jacobi-Poisson integrals to Jacobi-Poisson integrals is presented; in the third section, the characterization of Lipschitz functions defined...