数学物理学报(英文版) ›› 2005, Vol. 25 ›› Issue (2): 305-316.

• 论文 • 上一篇    下一篇

LOCAL INEQUALITIES FOR SIDON SUMS AND THEIR APPLICATIONS

范爱华,章逸平   

  1. Department of Mathematics, Wuhan University, Wuhan 430072, China
    LAMFA, CNRS UMR 6140, University of Picardie, 33 Rue Saint Leu, 80039 Amiens, FranceDepartment of Mathematics, Wuhan University, Wuhan 430072, China
  • 出版日期:2005-04-20 发布日期:2005-04-20

LOCAL INEQUALITIES FOR SIDON SUMS AND THEIR APPLICATIONS

 FAN Ai-Hua, ZHANG Yi-Beng   

  • Online:2005-04-20 Published:2005-04-20

摘要:

The authors consider Sidon sets of first kind. By comparing them with the
Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As
consequences, they prove the following results for Sidon series taking values in a Banach
space: the summability on a set of positive measure implies the almost everywhere con-
vergence; the contraction principle of Billard-Kahane remains true for Sidon series. As
applications, they extend a uniqueness theorem of Zygmund concerning lacunary Fourier
series and an analytic continuation theorem of Hadamard concerning lacunary Taylor se-
ries. Some of their results still hold for Sidon sets of second kind.

Abstract:

The authors consider Sidon sets of first kind. By comparing them with the
Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As
consequences, they prove the following results for Sidon series taking values in a Banach
space: the summability on a set of positive measure implies the almost everywhere con-
vergence; the contraction principle of Billard-Kahane remains true for Sidon series. As
applications, they extend a uniqueness theorem of Zygmund concerning lacunary Fourier
series and an analytic continuation theorem of Hadamard concerning lacunary Taylor se-
ries. Some of their results still hold for Sidon sets of second kind.

Key words: Sidon set, Khintchine-Kahane inequality, maximal inequality, comparison
principle,;contraction principle

中图分类号: 

  • 28A35