LOCAL INEQUALITIES FOR SIDON SUMS AND THEIR APPLICATIONS

Abstract

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

CLC Number:

28A35

FAN Ai-Hua, ZHANG Yi-Beng. LOCAL INEQUALITIES FOR SIDON SUMS AND THEIR APPLICATIONS[J].Acta mathematica scientia,Series B, 2005, 25(2): 305-316.

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