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Some Theorems on the Relation between Riesz and Abel Typical Means

Published online by Cambridge University Press:  24 October 2008

B. Kuttner
B. Kuttner
Affiliation:
The UniversityBirmingham
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Extract

Let {λn} be a sequence of non-negative numbers increasing to infinity. Let be any series. Following the usual terminology, we say that the series is summable (R, λn, k) to s if as u → ∞, where . If, further, u−kA(k)(u) is of bounded variation in (0, ∞), we say that the series is summable s. We say that the series is summable (A, λn) to s if converges for all σ > 0, and tends to s as σ →0+. If A, B are two summability methods, we write A → B (‘A implies B’) if they have the property that any series summable A is necessarily summable B to the same sum.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 1 , January 1961 , pp. 61 - 75
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

* Garabedian, H. L., Theorems associated with the Riesz and Dirichlet's series methods of summability, Bull. Amer. Math. Soc. 45 (1939), 891–5.CrossRefGoogle Scholar

† I am indebted to the referee for pointing this out.

‡ I use CM to denote Chandrasekharan, K. and Minakshisundaram, S., Typical means (Bombay, 1952).Google Scholar

* Widder, D. V., The Laplace transform (Princeton, 1941), 89 (Theorem 11·6b). In the theorem as stated by Widder, we put .Google Scholar

* To avoid repeated suffixes, we write λ(n) for λn whenever n is replaced by an expression which itself involves suffixes. We use a similar notation with other letters in place of λ.

† The notation given by (16) will be used throughout the paper.

* It is to ensure this that we have to take and not .

† Here we appeal to Theorem 1·61 of CM rather than to the modification given by Lemma 2.

‡ Here and elsewhere a ‘turning point’ of a function is to be taken to mean a point at which its derivative vanishes.

* These equations determine uniquely the ratio of the b's.

* This is possible since (except in the trivial case in which all the relevant a's vanish) is not identically zero for . For if it were, we would have and any one of equations (49) shows that we would then also have .

† In the case m = 0, (48) is to be replaced by (47), and (52) omitted.