Abstract
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.
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This work was supported by the National Natural Science Foundation of China (No. 10571122, 10971141), the Beijing Natural Science Foundation (No. 1052006, 1092004), the Project of Excellent Young Teachers and the Doctoral Programme Foundation of National Education Ministry of China, and the Project of Beijing Education Ministry.
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Li, Z., Song, F. & Li, K. Characterizations of Smoothness of Functions in Terms of their Jacobi Expansions. Results. Math. 59, 13–34 (2011). https://doi.org/10.1007/s00025-010-0047-z
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DOI: https://doi.org/10.1007/s00025-010-0047-z