Lacunary Trigonometric Series. Ii
نویسنده
چکیده
where E c [0, 1] is any given set o f positive measure and {ak} any given sequence of real numbers. This theorem was first proved by R. Salem and A. Zygmund in case of a -0, where {flk} satisfies the so-called Hadamard's gap condition (cf. [4], (5.5), pp. 264-268). In that case they also remarked that under the hypothesis (1.2) the condition (1.3) is necessary for the validity of (1.5) (cf. [4], (5.27), pp. 268-269). Further, in [2] P. Erdos has pointed out that for every positive constant c there exists a sequence of positive integers {nk} such that nk+l > nk(1 + ck-1/2), k>1, and (1.5) is not true for ak =1, k>1, and E = [0, 1]. But I could not follow his argument on the example. The purpose of the present note is to prove the following Theorem B. For any given constants c>0 and 0 + oo, are satisfied, but (1.5) is not true for E=[0, 1 ] and ak = 0, k>1. The above theorem shows that in Theorem A the condition (1.3) is
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