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The Borel property for simple Riesz means. (English) Zbl 0597.10051
Let \(P=(p_ n)\) be a sequence of positive real numbers. The complex sequence \((\alpha_ n)\) is called P-summable, if \((1/P_ N)\sum^{N}_{n=1}p_ n\alpha_ n\) converges, where \(P_ N=\sum^{N}_{n=1}p_ n\). Assuming that P is increasing, it is shown that almost all sequences (with respect to any probability measure on \({\mathbb{C}})\) are P-summable iff ”Hill’s condition” (H) holds, i.e. \(\sum^{\infty}_{n=1}\exp (-\delta /a_ n)<\infty\) for all \(\delta >0\), where \(a_ n=P_ n^{-2}\sum^{n}_{k=1}p^ 2_ k\). The sufficiency of (H) is due to J. D. Hill [Pac. J. Math. 1, 399-409 (1951; Zbl 0043.286)]. For arbitrary (non-monotonic) weights, (H) is still sufficient, but not necessary. We give a related necessary and sufficient condition.
MSC:
11K06 General theory of distribution modulo \(1\)
40D09 Structure of summability fields
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References:
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