Characterizations of Smoothness of Functions in Terms of their Jacobi Expansions

• DOI: 10.1007/s00025-010-0047-z
• Publication Date: 2010-07-06T06:21:33Z
• Authors: Zhongkai Li; Futao Song; Kai Li
• Keywords: Smoothness

The smoothness of functions defined on (−1, 1) is characterized by the Poisson integrals of their Jacobi expansions. The generalized translation T t and the generalized difference \({\widetilde{T}_t}\) associated with the product formulas of Jacobi polynomials are used to illustrate the smoothness of functions, which leads to very natural generalizations of the classical results of Hardy–Littlewoood and Zygmund.