On scales of summability methods

by Rüdiger Kiesel at University of Ulm

Abstract. In this paper we consider general Nörlund means (N_{αp}), α > −1 and power series methods (J_{p}) and the iteration product of two such methods. A particular case is that of the Cesàro methods (C_{α}) with corresponding power series method (A) , i.e. Abel’s method. We obtain generalizations of inclusion and tauberian and convexity theorems, which are well known for the Cesàro-Abel case, for a rich class of methods of the above type.

1 Introduction and main result

Let us start by recalling the following summability methods, see [8,15,20]: We say a sequence (s_{n}) of complex numbers is summable to s by the

(i) Cesàro-method s_{n} → s(C_{α}) for α > −1 if

σ_{n}^{α} := \frac{1}{\binom{n+α}{n}} ∑_{k=0}^{n} \binom{n − k + α − 1}{n − k}s_{k} → s (n → ∞),

(ii) Abel-method s_{n} → s(A) if

σ(x) := (1 − x) ∑_{n=0}^{∞} s_{n}x^{n} → s (x → 1−).

The Cesàro- and Abel-method are known to be closely related. Recall especially the following classical result, see [8, Theorem 92; 20, §53 I and §55 II, IV].

Theorem HL.

(i) For −1 < α ≤ β:

s_{n} → s(C_{α}) implies s_{n} → s(C_{β}) implies s_{n} → s(A). (Briefly: (C_{α}) ⊆ (C_{β}) ⊆ (A) ).

(ii) s_{n} → s(A) and s_{n} = O (1) imply s_{n} → s(C_{α}) for all α > 0 .

Statement (i) is an implication of Abelian character. It say, that the applicability of the Cesàro methods grows wider with increasing α, in the sense that the sets of summable sequences are increasing. For α → ∞ the set of (A) -summable sequences can be viewed as a limit set, although there are (A) -summable sequences which are not summable by any (C_{α}) -method.

Statement (ii) is of Tauberian character and really says that for bounded sequences the class of Cesàro methods and Abel’s method are equivalent (briefly (C_{α}) ≈ (A) ), if one imposes a Tauberian condition on the sequence (s_{n}) (in this case boundedness).

Theorem HL can be generalized to a so-called convexity theorem [1; 8 Theorem 70; 20 §53 V, §55]

Theorem A. For −1 < α < β ≤ γ:

(i) s_{n} → s(A) and s_{n} = O (1) (C_{α}) (i.e. σ_{n}^{α} = O (1) ) imply s_{n} → s(C_{β}).

(ii) s_{n} → s(C_{γ}) and s_{n} = O (1) (C_{α}) imply s_{n} → s(C_{β}) .

Using theorem HL we see that (ii) is an corollary of (i). In proving tauberian theorems it is also very useful to consider the iteration product of Cesàro and Abel methods [20, §55 V]. It is defined as follows, see [4,9]:

We say a sequence (s_{n}) of complex numbers is summable to s by the iteration product (AC_{α}) s_{n} → s(A, α) for α > −1 if

σ_{n}^{α} → s(A), i.e. (1 − x) ∑_{n=0}^{∞} σ_{n}^{α}x^{n} → s (x → 1−).

Observe that (A, 0) -summability is ordinary Abel-summability. (For summability methods (M) and (N) we always use the term iteration product (MN) meaning the (M) method applied to the (N) transform.) We have [9, Theorems 2.2, 2.3]

Theorem J. Let −1 < α ≤ β, −1 < γ. Then

(i) s_{n} → s(A, α) implies s_{n} → s(A, β).

(ii) s_{n} → s(C_{γ}) implies s_{n} → s(A, α) .

The aim of this paper is to discuss similar relations for a general class of power series methods and related matrix methods of summability. We consider general power series methods (J_{p}) defined in the following way [3,6,13,18]. Assume that (p_{n}) is a sequence of reals with

(1.1)  \{ \begin{matrix}p_{0} > 0, p_{n} ≥ 0, n ∈ N I , such that the power series \\ p(x) = ∑_{n=0}^{∞} p_{n}x^{n} has radius of convergence R = 1. \end{matrix}

For complex sequences (s_{n}) with increments a_{n} = s_{n} − s_{n−1}, n ≥ 0, s_{−1} = 0 , we consider the power series method of summability (J_{p}) , where we say

(1.2)  \{ \begin{matrix}s_{n} → s(J_{p}), & if p_{s}(x) = ∑_{n=0}^{∞} s_{n}p_{n}x^{n} converges for |x| < 1 \\ & and if σ_{p}(x) = \frac{p_{s}(x)}{p(x)} → s as x → 1 − . \end{matrix}

The power series methods (J_{p}) contain Abel’s method, p_{n} ≡ 1 , as a special case. We require the (J_{p}) -methods to be regular, i.e. s_{n} → s (n → ∞) implies s_{n} → s(J_{p}) . By Theorem 5 in [8], we have regularity if and only if

(1.3) P_{n} := ∑_{k=0}^{n} p_{k} → ∞, (n → ∞),

Without further mention we always assume (1.1) and (1.3).

For a general discussion the weights p_{n} cannot be too irregular, so we have to impose a regularity condition,

(1.4) np_{n} = O (P_{n}) , (n → ∞).

This includes especially the case of regularly varying weights p_{n} = n^{σ}L(n) , with σ ≥ −1 and L(.) an increasing, slowly varying function, i.e L ∈ SV , see [2], §1.2 for the definition.

As matrix methods, which have similar properties related to general power series methods as our paradigm Cesàro method we use, see [5,7,11,12], generalized Nörlund methods (N, a, b) .

The first definition of the generalized Nörlund transform (N, a, b) was given by D.Borwein in [5] and is as follows:

Let (a_{n}) and (b_{n}) be sequences of real numbers such that

(a ∗ b)_{n} := ∑_{k=0}^{n} a_{n−k}b_{k} ≠ 0, n = 0, 1, …

Given a sequence (s_{n}) of complex numbers, its sequence (σ_{n}^{a,b}) of (N, a, b) - transforms is defined by

(1.5) σ_{n}^{a,b} := \frac{1}{(a ∗ b)_{n}} ∑_{k=0}^{n} a_{n−k}b_{k}s_{k}

and we say s_{n} → s(N, a, b) , if σ_{n}^{a,b} → s (n → ∞) .

Using the Toeplitz theorem [8, Theorem 2] D.Borwein gave the necessary and sufficient conditions for the regularity of the (N, a, b) -methods,

(1.6)

(i) ∑_{k=0}^{n} |a_{n−k}b_{k}| = O((a ∗ b)_{n}) (n → ∞),
(ii) \frac{a_{n−k}b_{k}}{(a ∗ b)_{n}} → 0 (n → ∞) for each integer k.

We observe that the (C_{α}) -methods are special general Nörlund methods (N, ϵ^{α−1}, 1) obtained by using the sequences

ϵ^{α} = (ϵ_{n}^{α}) := (\binom{n + α}{n})

and 1 , where 1 stands for the sequence (1, 1, …) .

Let α > −1 . We consider the general Nörlund means (N, ϵ^{α−1}, p) , in short notation (N_{αp}) . So (C_{α}) is included as (N_{α1}) . We also use the notations:

p_{n}^{α} := (ϵ^{α−1} ∗ p)_{n} and σ_{n}^{α,p} := σ_{n}^{ϵ^{α−1},p}

for the (N_{αp}) transform.

We define an iteration product method (J_{p}, α) , α > −1 , where we say

s_{n} → a(J_{p}, α) if σ_{n}^{α,p} → s(J_{p}).

Our main results are

Theorem 1. Let (p_{n}) satisfy (1.4), then s_{n} → s(J_{p}) and s_{n} = O (1) imply s_{n} → s(N_{αp}) for all α > 0 .

Theorem 2. Let (p_{n}) satisfy (1.4). For −1 < α < β; s_{n} → s(J_{p}) and s_{n} = O (1) (N_{αp}) imply s_{n} → s(N_{βp}).

Theorem 3. Let p_{n} ∼ q_{n}, (n → ∞) , i.e. p_{n}/q_{n} → c > 0 (n → ∞) , with a sequence (q_{n}) that satisfies (1.4) and

(1.7) for each α ≥ 0 \frac{q_{n}}{q_{n}^{α}} is totally monotone

(see §2 for the definition). Then for bounded sequences (s_{n}) we have for 0 < α, β

(N_{αp}) ≈ (J_{p}, β).

We now start out in §2 to give the necessary definitions and a few preliminary results. In §3 we supply the proofs and conclude in §4 with a few remarks on related results and geralizations.

2 Definitions and preliminary results

For α, β ∈ R I the following identities are straightforward:

(2.1) p_{n}^{α+β} = ∑_{k=0}^{n} ϵ_{n−k}^{α−1}p_{k}^{β}, especially p_{n}^{α+1} = P_{n}^{α} = ∑_{k=0}^{n} p_{k}^{α}.

Remark 1. Using these identities we get for sequences (p_{n}) satisfying (1.4)

(i) P_{n}^{α} → ∞ for α ≥ 0 .

(ii) ϵ_{n}^{β−α−1} = o (p_{n}^{β}) , n → ∞ for 0 ≤ α < β.

(iii) (p_{n}^{α}) satisfies (1.4) for each α ≥ 0 .

(iv) \frac{p_{n}^{α}}{P_{n}^{α}} = O (\frac{p_{n}^{α+1}}{P_{n}^{α+1}} ) for α ≥ 0 .

Now we have the following inclusion property Proposition 1. For −1 < α ≤ β we have (N_{αp}) ⊆ (N_{βp}) ⊆ (J_{p}).

The propositon is proved via the two following lemmas. Lemma 1. If ((a ∗ p)_{n}) satisfies (1.1) and (1.3), then (N, a, p) ⊆ (J_{p}).

P r o o f: Let s_{n} → s (N, a, p) . By the regularity of all methods follows

σ_{p}(x) = \frac{p_{s}(x)}{p(x)} = \frac{∑_{n=0}^{∞}(a ∗ p)_{n} (\frac{1}{(a ∗ p)_{n}} ∑_{k=0}^{n} a_{n−k}p_{k}s_{k}) x^{n}}{∑_{n=0}^{∞}(a ∗ p)_{n}x^{n}} → s, (x → 1−).

Remark 2. The above proof shows that (J_{p}) ≈ (J_{p}^{α}, α) for each α ≥ 0 .

The (N_{αp}) methods satisfy a product formula. Lemma 2. For −1 < α ≤ β we have (N_{βp}) = (N, ϵ^{β−α−1}, p^{α})(N_{αp}). P r o o f: This follows from the identity

∑_{k=0}^{n} ϵ_{n−k}^{β−α−1} ∑_{ν=0}^{k} ϵ_{k−ν}^{α−1}p_{ν}s_{ν} = ∑_{ν=0}^{n} p_{ν}s_{ν} ∑_{k=ν}^{n} ϵ_{k−ν}^{α−1}ϵ_{n−k}^{β−α−1} = ∑_{ν=0}^{n} ϵ_{n−ν}^{β−1}p_{ν}s_{ν}.

P r o o f of Proposition 1: By Remark 1 (N, ϵ^{β−α−1}, p^{α}) (use (1.6)) and (J_{p^{α}}) are regular, so the Proposition follows from Lemmas 1 and 2.

As a further elementary result we compute the sequence-to-series form of the (N_{αp}) -transforms.

Lemma 3. For α > −1, n ∈ N I we have σ_{n}^{α,p}−σ_{n−1}^{α,p} = \frac{p_{n}^{α−1}}{p_{n}^{α}} (σ_{n}^{(α−1),p} − σ_{n−1}^{α,p}) .

P r o o f: Using the well know identities for the binomial coefficients and the identities (2.1) we obtain:

σ_{n}^{α,p} − σ_{n−1}^{α,p} = \frac{p_{n}s_{n}}{p_{n}^{α}} + \frac{1}{p_{n}^{α}p_{n−1}^{α}} ∑_{k=0}^{n−1} p_{k}s_{k} \{ϵ_{n−k}^{α−1}p_{n−1}^{α} − ϵ_{n−1−k}^{α−1}p_{n}^{α}\}

= \frac{p_{n}s_{n}}{p_{n}^{α}} + \frac{1}{p_{n}^{α}p_{n−1}^{α}} ∑_{k=0}^{n−1} p_{k}s_{k} \{p_{n−1}^{α}(ϵ_{n−k}^{α−1} − ϵ_{n−1−k}^{α−1}) − ϵ_{n−1−k}^{α−1}p_{n}^{α−1}\}

= \frac{p_{n}s_{n}}{p_{n}^{α}} + \frac{1}{p_{n}^{α}} ∑_{k=0}^{n−1} ϵ_{n−k}^{α−2}p_{k}s_{k} + \frac{p_{n}^{α−1}}{p_{n}^{α}p_{n−1}^{α}} ∑_{k=0}^{n−1} ϵ_{n−1−k}^{α−1}p_{k}s_{k}

= \frac{p_{n}^{α−1}}{p_{n}^{α}} \{\frac{1}{p_{n}^{α−1}} ∑_{k=0}^{n} ϵ_{n−k}^{α−2}p_{k}s_{k} − \frac{1}{p_{n−1}^{α}} ∑_{k=0}^{n−1} ϵ_{n−1−k}^{α−1}p_{k}s_{k}\} .

An immediate consequence is

Corollary 1. If s_{n} = O (1) and α ≥ 1 then |σ_{n}^{α,p} − σ_{n−1}^{α,p}| = O (\frac{p_{n}^{α−1}}{p_{n}^{α}} ) .

To discuss the (J_{p}, α) -methods we have to impose a further regularity condition. We need

Definition. A sequence (μ_{n}) is totally monotone, if there is a nondecreasing function χ : [0, 1] → R I such that

μ_{n} = ∫_{0}^{1} t^{n}dχ(t).

We then have

Proposition 2. Fix α ≥ 0 . If p_{n}/p_{n}^{α} is totally monotone, then (J_{p}) ⊆ (J_{p}, α) .

P r o o f: By Remark 2 we have (J_{p}) ⊆ (J_{p^{α}}, α) . Under the above condition the proposition follows from a theorem of D.Borwein in [3], see also [6,12].

We now state for convenience a few known results, which will be needed in the proofs. We start with a version of R.Sinha’s Theorem [16].

Theorem S. Let (p_{n}) satisfy (1.4), then for −1 < α < β ≤ γ

s_{n} → s(N_{γp}) and s_{n} = O (1) (N_{αp}) imply s_{n} → s(N_{βp}).

Under our conditions an alterative p r o o f of this theorem using Vitali’s Theorem [19, p.168] can be given, compare W.B.Jurkat [10]. Consider σ_{n}^{z,p} as an analytic function of the complex variable z = s + it in the rectangle R_{δ} := [α + δ ≤ s ≤ γ + 1, |t| ≤ δ] with a fixed, sufficiently small δ > 0 . Our assumptions imply that σ_{n}^{z,p} is uniformly bounded in R_{δ} and by Vitali’s Theorem the convergence of σ_{n}^{z,p} for γ ≤ s ≤ γ + 1 implies convergence for each inner point of R_{δ} .

The following two theorems are from H.Tietz/R.Trautner [18].

Theorem ST. Let (p_{n}) satisfy (1.4), then s_{n} = O (1) is a tauberian condition of type (J_{p}) → (M_{p}) , i.e. s_{n} → s(J_{p}) and s_{n} = O (1) imply s_{n} → s(M_{p}) .

Theorem TT. Let (p_{n}) satisfy (1.4), then a_{n} = O (\frac{p_{n}}{P_{n}} ) is a tauberian condition of type (J_{p}) to convergence.

3 Proofs

First p r o o f of Theorem 1. By Theorem ST we have s_{n} → s(J_{p}) and s_{n} = O (1) imply s_{n} → s(M_{p}) . Since (M_{p}) convergence is just (N_{1p}) convergence, the inclusion property gives s_{n} → s(N_{αp}) for α ≥ 1 . For 0 < α < 1 we choose a 0 < δ < α and then s_{n} = O (1) implies s_{n} = O (1) (N_{δp}) . Using Theorem S we obtain s_{n} → s(N_{αp}) .

Second p r o o f of Theorem 1. We start with a fixed β > 1 . By Remark 2 we see that s_{n} → s(J_{p^{β}}, β) . Using Corollary 1 and Remark 1 it follows that under s_{n} = O (1) the sequence (σ_{n}^{β,p}) satiesfies the tauberian conditions of Theorem TT. Hence we have s_{n} → s(N_{βp}) . Using the inclusion property and Theorem S as in the first proof the result follows.

Let p_{n}^{∗k} be the n− th coefficient of the power series p(x)^{k}, k ∈ N I . As a corollary we obtain a result, which was proved in a different setting in [11].

Corollary 2. If p_{n} ∼ n^{ρ}L(n) with ρ > −1 and L(.) a slowly varying function, then s_{n} → s(J_{p}) and s_{n} = O (1) imply s_{n} → s(N, p^{∗k}, p) for each k ∈ N I .

P r o o f: Theorem 1 gives s_{n} → s(N_{αp}) for each α > 0 . For α sufficiently small we use [7, Theorem 1 resp. Theorem 3] to get (N_{αp}) ⊆ (N, p, p) . The inclusion property [11, Proposition 1] for the (N, p^{∗k}, p) -methods then gives the result.

First p r o o f of Theorem 2. Using again Remark 2 we get s_{n} → s(J_{p^{α}}, α) . Since σ_{n}^{α,p} = O (1) we obtain by Theorem ST σ_{n}^{α,p} → s(M_{p^{α}}) . By the product formula this is s_{n} → s(N_{(α+1)p}) . By inclusion and Sinha’s Theorem the result follows.

Second p r o o f of Theorem 2. Let α < β. As above we have s_{n} → s(J_{p^{β+1}}, β + 1) . Using Lemma 2 we

see that this means σ_{n}^{α,p} → s(J_{p^{β+1}})(N_{(β+1−α)p^{α}}) . Write σ̃_{n}^{(β+1−α),p^{α}} for the (N_{(β+1−α)p^{α}}) -transform of (σ_{n}^{α,p}) . Since σ_{n}^{α,p} = O (1) it follows by Corollary 1 and Remark 1 (iv) that

σ̃_{n}^{(β+1−α),p^{α}} − σ̃_{n−1}^{(β+1−α),p^{α}} = O (\frac{p_{n}^{β+1}}{P_{n}^{β+1}} ) .

Hence (σ̃_{n}^{(β+1−α),p^{α}}) satisfies the tauberian conditions of Theorem TT, and we obtain σ̃_{n}^{(β+1−α),p^{α}} → s. By the product formula this is s_{n} → s(N_{(β+1)p}) , and the inclusion property and Theorem S give the desired result.

P r o o f of Theorem 3. In view of Theorem 1 and Proposition 1 it is only left to prove that for bounded sequences (s_{n}) (J_{p}, β) ≈ (J_{p}) for each β > 0 . We start by proving that s_{n} = O (1) is a tauberian condition for (J_{p}, β) → (J_{p}) for each β > 0 .

To this end we show the stronger proposition that s_{n} = O (1) is a tauberian condition for (J_{p}, β) → (N_{γp}) for each γ > 0 . Since s_{n} = O (1) we have σ_{n}^{β,p} = O (1) and we can switch to the asymptotic weights (compare [13, Lemma 2]) this means σ_{n}^{β,p} → s (J_{q}) . By Theorem ST σ_{n}^{β,p} → s(M_{q}) . Now (1.7) implies that q_{n}^{β}/q_{n} is nondecreasing and by using (1.4) we get

\frac{q_{n}^{β}Q_{n}}{Q_{n}^{β}q_{n}} = O (1) .

By [8, Theorem 14] these conditions imply that σ_{n}^{β,p} → s(M_{q^{β}}) . Now again the boundedness of σ_{n}^{β,p} allows us to switch back to the weights (p_{n}) and obtain by the product property s_{n} → s(N_{(β+1)p}) . Since under s_{n} = O (1) all (N_{γp}) - methods are equivalent for γ > 0 the desired result follows. The inclusion (J_{p}) ⊆ (J_{p}, β) for each β > 0 follows by using the asymptotic weights, argue as above, and by Proposition 2.

4 Concluding Remarks

(i) Sinha’s original theorem is more general and reads as follows: Let (q_{n}) be a sequence of constants such that for each ξ > −1 there exist positive constants H_{1}, H_{2} (which may depend on ξ but not on n) with

H_{1}n^{ξ} ≤ (p^{ξ} ∗ q)_{n}/(p ∗ q)_{n} ≤ H_{2}n^{ξ}.

Then if −1 < α < β ≤ γ, s_{n} → s(N, p^{γ}, q) and s_{n} = O (1) (N, p^{α}, q) imply s_{n} → s(N, p^{β}, q) .

If (q_{n}) satisfies the conditions (1.1) and (1.3) and

p_{n}^{ξ} = o (n^{ξ}(p ∗ q)_{n}) (n → ∞),

for each ξ ≥ 0 , then one can prove theorems similar to Theorems 1-3 (using the first method of proof, since a sequence-to-series form is not accessible in closed form).

Theorem 1’. s_{n} → s(J_{q}) and s_{n} = O (1) imply s_{n} → s(N, p^{α}, q) for all α > 0 .

Theorem 2’. For −1 < α < β:

s_{n} → s(J_{q}) and s_{n} = O (1) (N, p^{α}, q) imply s_{n} → s(N, p^{β}, q).

Theorem 3’. Let

\frac{p_{n}}{(p^{α} ∗ q)_{n}} ∼ μ_{n}(α), (n → ∞)

with (μ_{n}(α)) totally monotone for each α ≥ 0 . Then for bounded sequences (s_{n}) we have for 0 < α, β

(N, p^{α}, q) ≈ (J_{q}, p^{β}),

where (J_{q}, p^{β}) denotes the iteration product of the (J_{q}) and (N, p^{β}, q) methods.

In this scale the (M_{q}) -methods are not directly included, but one can show (with Theorem 1 in [7]) that (M_{q}) ⊆ (N, p^{α}, q) for each α > 0 .

(ii) In [17] A.Tali discusses further convex families of general Nörlund transforms. These methods could likewise be used to construct scales of summability methods including power series methods.

(iii) Lemma 3 also allows a further application of the so-called modified reduction method introduced by M.R.Parameswaran in [14].

Proposition 3. Let (p_{n}) satisfy (1.4) and (V) be a summability method such that

(a) na_{n} = O (1) is a tauberian condition for (V) to convergence for bounded sequences.

(b) (V) ⊆ (V N_{αp}) for some α ≥ 1 .

Then (V) ⊆ (N_{αp}) for bounded sequences.

P r o o f: s_{n} → s(V) implies by (b) s_{n} → s(V N_{αp}) . Under s_{n} = O (1) by Lemma 3 and (1.4) (σ_{n}^{α,p} − σ_{n−1}^{α,p})n = O (1) . Hence by (a) s_{n} → s(N_{αp}) .

(iv) For the case p_{n} = \frac{1}{n+1} , i.e the logarithmic method (L) , s_{n} = O (1) is not a tauberian condition of type (J_{p}) → (N, p, p) (compare Corollary 2). This can be seen as follows.

Set L(n) := ∑_{k=0}^{n} \frac{1}{k + 1} and

τ_{n} := \frac{1}{L(n)} ∑_{k=0}^{n} \frac{1}{n − k + 1}s_{k}  the Nörlund (harmonic) mean (N, \frac{1}{n+1}),
t_{n} := \frac{1}{L(n)} ∑_{k=0}^{n} \frac{1}{k + 1}s_{k}  the weighted (logarithmic) mean (l),
σ_{n} := \frac{n + 2}{2L(n)} ∑_{k=0}^{n} \frac{1}{n − k + 1} \frac{s_{k}}{k + 1} the generalized Nörlund mean (N, \frac{1}{n+1}, \frac{1}{n+1}).

Straightforward computation show that τ_{n} + t_{n} = 2σ_{n}. Hence whenever two of the above methods sum a sequence so does the third. It is also known [use e.g. 8, Theorem 23] that (N, \frac{1}{n+1}) ⊆ (C_{1}) . From [8, §3.8 and Theorem 82] we know that there is a bounded sequence (A_{n}) which is summable (l) but not (C_{1}) . Now if boundedness were a tauberian condition, summability of (A_{n}) by (l) would imply summability (N, \frac{1}{n+1}, \frac{1}{n+1}) and via (N, \frac{1}{n+1}) summability (C_{1}) a contradiction.

List of References

1. Andersen, A.F.: Studier over Cesáro’s Summabilitetsmetode, Dissertation, Köbenhavn (1921)

2. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, (Encycl.- Math.Appl. 27), Cambridge University Press, 1987

3. Borwein, D.: On summability methods based on power series, Proc.Royal-Soc.Edinburgh Sect.A 64 (1957), 342–349

4. Borwein, D.: Theorems on some methods of summability, Quart.J.Math.- Oxford(2) 9 (1958), 310–316

5. Borwein, D.: On products of sequences, J.London Math.Soc.33 (1958), 352–357

6. Baron, S.; Tietz, H.: Produktsätze für Potenzreihenverfahren und verallgemeinerte Nörlund Mittel, Tartu Ülikooli Toimetised, Functional analysis and theory of summability 960, (1993), 13–22

7. Das, G.: On some methods of summability II, Quart.J.Math.Oxf.(2)19 (1968), 417–431

8. Hardy, G.H.: Divergent Series, (Oxford: University Press, 1949)

9. Jakimovski, A.: On a converse of Abel’s theorem, Proc.Amer.Math.Soc. 3 (1952), 244–252

10. Jurkart, W.B.: Ein funktionentheoretischer Beweis für O-Taubersätze bei Potenzreihen, Archiv Math. 7, (1956), 122–125

11. Kiesel, R.: General Nörlund transforms and power series methods, M.Z. 214, 273-286, (1993)

12. Kiesel, R.; Stadtmüller, U.: Tauberian- and Convexity theorems for certain (N, p, q) -means, to appear in Can.Math.Bull.

13. Kratz, W., Stadtmüller, U.: O–Tauberian Theorems for (J_{p})-methods with rapidly increasing weights, J.London Math.Soc. (2) 41 (1990), 489–502

14. Parameswaran, M.P.: On a Generalization of a theorem of Meyer-König, M.Z. 162 (1978), 201–204

15. Peyerimhoff, A.: Lectures on Summability, Lecture Notes in Mathematics, No.107, (Springer Verlag 1969)

16. Sinha, R.: Convexity theorems for (N, p, q) Summability, Kyungpook Math.J. 13 (1973), 37–40

17. Tali, A.: Convexity theorems for families of summability methods, Tartu Ülikooli Toimetised, Functional analysis and theory of summability 960, (1993), 117–138

18. Tietz, H., Trautner, R.: Taubersätze für Potenzreihen, Archiv Math. 50, (1988), 164–174

19. Titchmarsh, E.C.: The theory of functions, (Oxford 1939)

20. Zeller K., Beekmann, W.: Theorie der Limitierungsverfahren, (Erg.- Math.Grenzgebiete, 15), (Springer Verlag, 1970)

Universität Ulm Abteilung Stochastik Helmhotzstr. 18 89069 Ulm Germany email: kiesel@mathematik.uni-ulm.de