A Unified Approach to Uniqueness of Walsh Series and Haar Series

Abstract

ABSTRACT. We obtain a uniqueness theorem for Walsh series valid for subsequences of 2nth partial sums which satisfy a pointwise growth condition. Introduction.There are three techniques for obtaining uniqueness results for Walsh series: quasimeasures, formal integration, and nested intervals.The quasimeasure technique, introduced by Yoneda (see [8]), is designed to obtain uniqueness results for Walsh series which are everywhere finite but satisfy no other growth condition.The formal integral technique, originated by Riemann for trigonometric series, adapted for Walsh series by Fine [4], and developed by many others (e.g., Crittenden and Shapiro [3], and Lindahl [5]) obtains uniqueness results for Walsh series satisfying pointwise growth conditions, but does not apply to subsequences of 2"th partial sums.