Acta Mathematica Academiae Scientiarum Hungaricae 61. (1993)

1993 / 1-2. szám - Yoneda, K.: Uniqueness theorems for Walsh series under a strong condition

UNIQUENESS THEOREMS FOR WALSH SERIES UNDER A STRONG CONDITION K. YONEDA (Osaka) Let 1. Introduction OO p = ^2Mk)wk(x) к=0 be a Walsh series and A a certain class of Walsh series. When E is a subset of the dyadic group, it is called a set of uniqueness for A, if p € A and 2" —1 lim У fi(k)wk(x) = 0 everywhere except on E k=0 imply that p(k) = 0 for к = 0,1,2,__When E is not a set of uniqueness for A, it is called a set of multiplicity for A. It is easy to see that a subset of the dyadic group is a set of uniqueness for the class of all Walsh series ft such that X/ IM^OI2 < k=o if and only if it is of measure zero. This class of Walsh series coincides with L2 -space. In this paper we shall consider the uniqueness problem for the class of all Walsh series p such that 2n+l—1 Kk)wk(x) k-2" 2"|Дт^(/„(х))| = o(l) uniformly in x as n —► oo. Let В be the class of these Walsh series. It is easy to see that В and L2 are not subsets of each other. We shall prove the following three theorems. THEOREM 1. Assume that E is a subset of the dyadic group, and there exists a couple of sequences of integers {iVn}„ = {Nn(x)}n and {kn}n = {fcn(x)}„ for each x 6 E such that (i) Nn T 00 as n —> oo; Acta Math. Hung. 61 (1-2) (1993), 1-15.

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