ABSOLUTE φ -SUMMABILITY FACTORS WITH A POWER FOR A^{α} -METHODS ^{∗}

Simson Baron and Rüdiger Kiesel

Received:

ABSTRACT. We give necessary and sufficient conditions for a sequence (ε_{n}) of complex numbers to be summability factors of type (φ−|A|_{r} , φ−|B|_{s}) for various (r, s) combinations, where A is an A^{α} matrix of summability and B is a triangular matrix of summability satisfying certain regularity conditions. As examples we discuss summability factors of the above type in case B = Z_{β}, the method of Silverman-Szasz, and in case A = M_{a} and B = M_{b} with M_{a} and M_{b} being summability methods of weighted means.

AMS subject classification (1991): 40C05, 40D15, 40F05

1 Introduction and main result

Let φ = (φ_{n}) be a sequence of positive numbers and

∑ u_{n}  (1)

be a series of complex numbers u_{n}. Unless otherwise indicated free indices always run through all values 0, 1, 2, …, and summation is likewise over 0, 1, 2, . . .. Let A be a triangular method of summability with the matrix (α_{nk}) of series-to-sequence transformation and with the matrix (ᾱ_{nk}) of series-to-series transformation. We set

v_{n} = ∑_{k=0}^{n} ᾱ_{nk}u_{k}.  (2)

∗This research was enabled by the generous support of the Minerva Foundation in Germany at the Emmy-Noether-Institute at Bar-Ilan University.

2 Simson Baron and Rüdiger Kiesel The series (1) is said to be φ -absolute A -summable with the power p ≥ 1 or briefly, φ−|A|_{p} - summable, if (compare [7, 20])

∑ φ_{n}^{p−1} |v_{n}|^{p} < ∞.  (3)

Let B be a triangular method of summability, where by (β_{nk}) we denote the matrix of seriesto-sequence transformation and by (β̄_{nk}) the matrix of the series-to-series transformation. Let ε = (ε_{n}) be a sequence of complex numbers, and r, s ≥ 1 . The numbers ε_{n} are called summability factors of type (φ−|A|_{r} , φ−|B|_{s}) , if for any φ−|A|_{r} -summable series (1) the series

∑ ε_{n}u_{n}  (4)

is φ−|B|_{s} -summable, that is

w_{n} := ∑_{k=0}^{n} β̄_{nk}ε_{k}u_{k}  (5)

yields

∑ φ_{n}^{s−1} |w_{n}|^{s} < ∞

whenever (u_{n}) satisfies (3) (with p = r in this case). We will use in the following the short notation: ε ∈ (φ−|A|_{r} , φ−|B|_{s}) , where we omit the index if it is 1 . If B = E is the convergence method, then summability factors are called convergence factors of the corresponding type. Problems of convergence factors were discussed for a general class of matrices in [5]. In this note we discuss sufficient and necessary conditions so that ε ∈ (φ−|A|_{r} , φ−|B|_{s}) for the (r, s) combinations (p, 1); (1, p); (p, p) for a real number p > 1 and for the class of so-called A^{α} -matrices, which are defined as follows [1, 2]: Let α be a nonnegative integer. An A^{α} -method is called a normal method, whose series-to-series matrix (ᾱ_{nk}) has an inverse matrix (η̄_{nk}) with exactly α + 1 non-zero diagonals, namely η̄_{nk} = 0 for n < k and n > k + α. The inverse matrix (η_{nk}) of the correspondig series-to-sequence matrix (α_{nk}) then satisfies η_{nk} = η̄_{nk} − η̄_{n,k+1} and has α + 2 non-zero diagonals.

Examples of A^{α} -methods:

(a) The Cesàro-methods C_{γ} of order γ = 0, 1, 2 … are A^{γ} -methods (see [3], p.86).

(b) The weighted means method M_{a} generated by a sequence a = (a_{n}) with a_{n} ≠ 0 is an A^{1} -method and the iteration product of two such methods is an A^{2} -method (see [2], p.177).

(c) The Nörlund-methods N_{a} with generating function

a(z) = ∑ a_{n}z^{n}

which has a polynomial reciprocal function

c(z) := \frac{1}{a(z)} = ∑_{k=0}^{α} c_{k}z^{k}  (6)

are A^{α} -methods, since η_{nk} = A_{k}(c_{n−k} −c_{n−k−1}) , where A_{n} := a_{0} +… a_{n} (see [3], p.103-104, and [16], p.17).

(d) Generalized Nörlund methods (N, a, b) are likewise A^{α} -methods if (6) is true (see [9, 11]).

In this paper we consider all the above mentioned types of summability factors, where A is an A^{α} -method and B satisfies some regularity conditions. Before stating our results we need a few notations. For the method A denote

D̄_{n} := sup_{k} |α_{n+k,n+k}η̄_{n+k,k}| , G_{k} := ∑_{ν=k}^{k+α} η̄_{νk}ε_{ν}.

Absolute φ-summability factors with a power 3 For the method B denote (cf. [3], p.52)

\begin{matrix}b_{nk} := Δβ_{nk} := β_{nk} − β_{n,k+1}, & b̄_{nk} := Δβ̄_{nk}, \\ b(k, p) := ∑_{n=k}^{∞} φ_{n}^{p−1} |b̄_{nk}|^{p} , & β(k, p) := ∑_{n=k}^{∞} φ_{n}^{p−1} |β̄_{nk}|^{p} . \end{matrix}

As usual the number q is defined by the equation 1/p + 1/q = 1 . We also use Landau’s o(.) and O(.) symbols, where the relevant index is supposed to tend to infinity. Let p > 1 . We always assume the existence of finite D̄_{n} at n = 1, 2, … α for the A^{α} -method and that B is a normal method (see [16], p.14).

THEOREM 1. Assume that

φ_{n+1} = O(φ_{n})  (7)

and let the method B satisfy the conditions

lim_{n→∞} β_{nk} = 1,  (8)
β_{kk} = O(β_{k+1,k+1}),  (9)
β(k, 1) = O(1),  (10)
b(k, 1) = O(β_{kk}).  (11)

Then ε ∈ (φ−|A^{α}|_{p} , |B|) if and only if the following conditions are satisfied:

∑ φ_{n}^{−1} |G_{n}|^{q} < ∞,  (12)
∑ φ_{n}^{−1} |\frac{β_{nn}ε_{nn}}{α_{nn}} |^{q} < ∞.  (13)

Remark 1. The condition (8) is fulfilled if B is a regular matrix. By the Knopp-Lorentz theorem (see [3], p.34; [16], p.13) B preserves absolute convergence if and only if (10) is satisfied.

Examples of matrices B which satisfy (8)-(11) are

(a) Cesàro-methods C_{β} with 0 ≤ β ≤ 1 (see [3], pp.85 and 188).

(b) M_{a} methods which preserve absolute convergence and are absolute translative from the right (see [3], pp.114 and 123; [2], p.176).

(c) N_{a} methods with decreasing, non-negative generating sequences a = (a_{n}) (see [15], Corollary 1; [2], p.175).

(d) Generalized Nörlund methods (N, a, b) (see [9, 11]), where the generating sequence a is also assumed to be decreasing.

THEOREM 2. Assume that φ satiesfies (7) and

φ_{n} = O(φ_{n+1})  (14)

and let the method B satisfy the conditions (9),

β_{k+1,k+1} = O(β_{kk}),  (15)
b(k, p) = O(φ_{k}^{p−1} |β_{kk}|^{p}).  (16)

Then ε ∈ (|A^{α}| , φ−|B|_{p}) if and only if the following conditions

β(k, p) |G_{k}|^{p} = O(1),  (17)
β_{kk}ε_{k} = O(φ_{k}^{−1/q}α_{kk})  (18)

are satisfied.

Examples of matrices B which fulfill (9), (15), (16):

(a) For C_{β} and φ_{n} = n + 1 the conditions are satisfied when −1 < β < 2 − 1/p (see [14], Lemma 5).

(b) M_{a} -means with regulary varying weights sequences a = (a_{n}) and φ a sufficiently smooth sequence (compare §6).

(c) (N, a, b) methods with generating sequences a_{n} = \binom{n+α}{n}, α ≥ 0 , and b_{n}n = O(B_{n}) (compare [11]).

For ε ∈ (φ−|A^{α}|_{p} , φ−|B|_{p}) we give sufficient conditions in our third theorem.

THEOREM 3. Assume that φ satisfies (7),(14) and that the method B satisfies the conditions (9), (15),

∑_{k=0}^{n} φ_{k}^{−1/q} |\frac{b̄_{nk}}{β_{kk}} | = O(φ_{n}^{−1/q}),  (19)

∑_{n=k}^{∞} φ_{n}^{1/q} |b̄_{nk}| = O(β_{kk}φ_{k}^{1/q}).  (20)

Then ε ∈ (φ−|A^{α}|_{p} , φ−|B|_{p}) if

∑_{k=0}^{n} φ_{k}^{−1/q} |β̄_{nk}G_{k}| = O(φ_{n}^{−1/q}),  (21)

∑_{n=k}^{∞} φ_{n}^{1/q} |β̄_{nk}G_{k}| = O(φ_{k}^{1/q}),  (22)
β_{kk}ε_{k} = O(α_{kk})  (23)

are satisfied.

In the following three paragraphs we prove Theorems 1-3 by using sufficient and necessary conditions for matrix transforms from l^{s} into l^{r} as given in [8, 10, 12, 17]. Our main tool is the special structure of the A^{α} matrices. To show how useful this matrices are we state and prove a further Theorem in §5, giving sufficient conditions for ε ∈ (φ−|A^{α}|_{p} , φ−|B|_{p}) . The type of these conditions has recently found wide use, see for example [7, 20, 21, 22] and the references therein. It turns out that by using our technique many of these results can also be obtained. As examples we provide summability factors for weighted means methods and the Silverman-Szazs methods.

Absolute φ -summability factors with a power 5

2 Proof of Theorem 1

Set x_{n} := φ_{n}^{1/q}v_{n} and w_{n} as in (5). Then using the inverse transformation of A we have

u_{k} = ∑_{ν=0}^{k} η̄_{kν}v_{ν} = ∑_{ν=0}^{k} η̄_{kν}φ_{ν}^{−1/q}x_{ν}.

Therefore

w_{n} = ∑_{ν=0}^{n} β̄_{nν}ε_{ν} ∑_{k=0}^{ν} η̄_{νk}φ_{k}^{−1/q}x_{k} = ∑_{k=0}^{n} x_{k}φ_{k}^{−1/q} ∑_{ν=k}^{n} β̄_{nν}ε_{ν}η̄_{νk},

and consequently

w_{n} = ∑_{k=0}^{n} φ_{k}^{−1/q}γ̄_{nk}x_{k},

where

γ̄_{nk} := ∑_{ν=k}^{n} β̄_{nν}ε_{ν}η̄_{νk}.

Now we can use the necessary and sufficient conditions for matrices transforming l^{p} into l^{1} (see [17], Theorem C). These conditions imply that ε ∈ (φ−|A^{α}|_{p} , |B|) if and only if

∑_{k=0}^{∞} φ_{k}^{−1} | ∑_{n∈K} γ̄_{nk}|^{q} = O(1)  (24)

for each K ∈ K, where K denotes the set of finite sets of distinct non-negative integers. Now (24) implies the necessity of

∑_{k=0}^{l} φ_{k}^{−1} |∑_{n=0}^{j} γ̄_{nk}|^{q} = O(1) ∀l, j = 0, 1, ….  (25)

Since γ̄_{nk} = 0 for k > n, we have for sufficiently large j ≥ k + α + 1 (see [3], p.51)

\begin{aligned}∑_{n=0}^{j} γ̄_{nk} & = & ∑_{n=k}^{j} ∑_{ν=k}^{n} β̄_{nν}ε_{ν}η̄_{νk} = ∑_{ν=k}^{j} ε_{ν}η̄_{νk} ∑_{n=ν}^{j} β̄_{nν} \\ & = & ∑_{ν=k}^{j} ε_{ν}η̄_{νk}β_{jν} = ∑_{ν=k}^{k+α} ε_{ν}η̄_{νk}β_{jν} → G_{k} (j → ∞) \end{aligned}

by (8). Hence (25) implies (12). Furthermore we obtain (13) directly from (24) via Mehdi’s Lemma (see [13], p.136, Lemma 12), which implies

∑ φ_{n}^{−1} \left|{ \atop γ̄_{nn}}\right|^{q} < ∞

and this is just (13) as

γ̄_{nn} = \frac{β̄_{nn}ε_{n}}{α_{nn}} .

To prove the sufficiency of (12) and (13) we show that these conditions yield

∑_{k=0}^{∞} φ_{k}^{−1} ∑_{n=k}^{∞} \left|{ \atop γ̄_{nk}}\right|^{q} < ∞,  (26)

which implies (24) because (γ̄_{nk}) is a triangular matrix. By partial summation we obtain

\begin{aligned}γ̄_{nk} & = & (∑_{ν=k}^{n} ε_{ν}η̄_{νk}) β̄_{n,n+1} + ∑_{ν=k}^{n} b̄_{nν} ∑_{μ=k}^{ν} ε_{μ}η̄_{μk} \\ & = & ∑_{ν=k}^{n} b̄_{nν} (∑_{μ=k}^{k+α} − ∑_{μ=ν+1}^{k+α}) ε_{μ}η̄_{μk} = ∑_{ν=k}^{n} b̄_{nν}G_{k} − ∑_{ν=k}^{n} b̄_{nν} ∑_{μ=ν+1}^{k+α} ε_{μ}η̄_{μk}. \end{aligned}

Hence we have

γ̄_{nk} = β̄_{nk}G_{k} − λ̄_{nk},  (27)

where

λ̄_{nk} = ∑_{ν=k}^{N} b̄_{nν} ( ∑_{μ=ν+1}^{k+α} ε_{μ}η̄_{μk}) , N := min{n, k + α − 1}.  (28)

Since B satisfies (10) the condition (12) yields

∑_{k=0}^{∞} φ_{k}^{−1} (∑_{n=k}^{∞} |β̄_{nk}G_{k}|)^{q} = ∑_{k=0}^{∞} φ_{k}^{−1} |G_{k}|^{q} β^{q}(k, 1) = O(1) ∑_{k=0}^{∞} φ_{k}^{−1} |G_{k}|^{q} < ∞.

In view of the latter relation, (27) and Minkowski’s inequality, it remains to prove that

∑_{k=0}^{∞} φ_{k}^{−1} (∑_{n=k}^{∞} |λ̄_{nk}|)^{q} < ∞.

Since B is triangular and satisfies (11) we have

∑_{ν=k}^{k+α} ∑_{n=ν}^{∞} |b̄_{nν}| = ∑_{ν=k}^{k+α} O (β_{νν})

and therefore by (9)

∑_{n=k}^{∞} |λ̄_{nk}| = O(1) ∑_{μ=k}^{k+α} |ε_{μ}η̄_{μk}β_{μμ}| = O(1) ∑_{μ=0}^{α} D̄_{μ} |\frac{ε_{k+μ}β_{k+μ,k+μ}}{α_{k+μ,k+μ}} | .  (29)

Hence by (7) and condition (13)

∑_{k=0}^{∞} φ_{k}^{−1} (∑_{n=k}^{∞} |λ̄_{nk}|)^{q} = O(1) ∑_{μ=0}^{α} ∑_{k=μ}^{∞} φ_{k}^{−1} |\frac{ε_{k}β_{kk}}{α_{kk}} |^{q} < ∞.

Remark 2. For φ_{n} = n + 1 this theorem was proved in [4]. For discussion of special methods and applications we also refer to this paper.

3 Proof of Theorem 2

We set y_{n} = φ_{n}^{1/q}w_{n} then it follows, recall (5)

y_{n} = φ_{n}^{1/q} ∑_{k=0}^{n} γ̄_{nk}v_{k}.  (30)

Absolute φ-summability factors with a power 7 By applying the theorem of Cohen-Dunford (see [8], p.696, Theorem 10, or [10], p.349, Corollary 1.2, or [12], p.167, Theorem 5) to the transformation (30) we obtain that ε ∈ (|A^{α}| , φ−|B|_{p}) if and only if

sup_{k} ∑_{n=k}^{∞} φ_{n}^{p−1} \left|{ \atop γ̄_{nk}}\right|^{p} < ∞.  (31)

By setting n = k in (31) we deduce

sup_{k} φ_{k}^{p−1} \left|{ \atop γ̄_{kk}}\right|^{p} = sup_{k} φ_{k}^{p−1} |β̄_{kk}ε_{k}η̄_{kk}|^{p} < ∞.

Hence we immediatly obtain (18). We postpone to show the necessity of (17) and turn to the sufficiency part. Recall the decomposition formula (27). We start by estimating |λ̄_{nk}|^{p} with the help of Hölders formula and (28) getting:

|λ̄_{nk}|^{p} ≤ (∑_{ν=k}^{N} |b̄_{nν}|^{p}) λ^{p/q}(k),  (32)

where

λ(k) := ∑_{ν=k}^{N} | ∑_{μ=ν+1}^{k+α} η̄_{μk}ε_{μ}|^{q} .

From (18), (14), (9) we get

λ(k) = O(1) ∑_{μ=0}^{α} D̄_{μ} |ε_{μ+k}/α_{μ+k,μ+k}|^{q} = O(φ_{k}^{−1} |β_{kk}|^{−q}).

Since B is normal and satisfies (15) and (16) we obtain by (7)

∑_{n=k}^{∞} φ_{n}^{p−1} ∑_{ν=k}^{N} |b̄_{nν}|^{p} ≤ ∑_{ν=k}^{k+α−1} b(ν, p) = O(φ_{k}^{p−1} |β_{kk}|^{p}).

By combining these estimates with (32) we get

∑_{n=k}^{∞} φ_{n}^{p−1} |λ̄_{nk}|^{p} = O(φ_{k}^{−p/q} |β_{kk}|^{−p} φ_{k}^{p−1} |β_{kk}|^{p}) = O(1).  (33)

From (17) and (33) we deduce (31) by (27), since |a + b|^{p} ≤ 2^{p}(|a|^{p} + |b|^{p}) for any real numbers a and b. The necessity of (17) now follows since the decomposition formula (27) implies

β̄_{nk}G_{k} = λ̄_{nk} + γ̄_{nk}

and by Minkowski’s inequality we obtain (17) vice versa from (31) and (33).

Remark 3. For φ_{n} = n + 1 this Theorem was proved in [4].

4 Proof of Theorem 3

As in §2 resp. §3 we set x_{n} = φ_{n}^{1/q}v_{n} resp. y_{n} = φ_{n}^{1/q}w_{n}. It follows from (30) that

y_{n} = ∑_{k=0}^{n} (\frac{φ_{n}}{φ_{k}} )^{1/q} γ̄_{nk}x_{k}.  (34)

To prove that the transformation (34) preserves l^{p} -convergence, we have to show (see [12], p.174, Theorem 9) that

∑_{k=0}^{n} φ_{k}^{−1/q} \left|{ \atop γ̄_{nk}}\right| = O(φ_{n}^{−1/q}),  (35)

∑_{n=k}^{∞} φ_{n}^{1/q} \left|{ \atop γ̄_{nk}}\right| = O(φ_{k}^{1/q}).  (36)

We start by proving (35). In view of the decomposition formula (27) and (21) it is only left to show that

∑_{k=0}^{n} φ_{k}^{−1/q} |λ̄_{nk}| = O(φ_{n}^{−1/q}).  (37)

By using (28) we get

|λ̄_{nk}| ≤ ∑_{μ=k}^{k+α−1} |b̄_{nμ}| ∑_{ν=μ+1}^{k+α} |ε_{ν}η̄_{νk}| = ∑_{ν=k+1}^{k+α} |ε_{ν}η̄_{νk}| ∑_{μ=k}^{ν−1} |b̄_{nμ}|

and by (23), (9), (15), (7) we obtain, since B is normal,

\begin{aligned}∑_{k=0}^{n} φ_{k}^{−1/q} |λ̄_{nk}| & = & O(1) ∑_{k=0}^{n} φ_{k}^{−1/q} ∑_{ν=1}^{α} D̄_{ν} ∑_{μ=k}^{ν+k−1} |\frac{b̄_{nμ}}{β_{k+ν,k+ν}} | \\ & = & O(1) ∑_{k=0}^{n} φ_{k}^{−1/q} |\frac{b̄_{nk}}{β_{kk}} | . \end{aligned}

Now (37) follows from (19). To prove (36) in the view of (22) and (27) it only remains to show that

∑_{n=k}^{∞} φ_{n}^{1/q} |λ̄_{nk}| = O(φ_{k}^{1/q}).  (38)

Now by (20), (9), (14), (23), (7) we get (38), in the same way as we obtained (29), namely

∑_{n=k}^{∞} φ_{n}^{1/q} |λ̄_{nk}| = O(1) ∑_{μ=k+1}^{k+α} |η̄_{μk}ε_{μ}| ∑_{ν=k}^{μ−1} ∑_{n=ν}^{∞} φ_{n}^{1/q} |b̄_{nν}|

= O(1) ∑_{μ=k+1}^{k+α} |η̄_{μk}ε_{μ}b_{μμ}| φ_{μ}^{1/q}

= O(1) ∑_{μ=k+1}^{k+α} |η̄_{μk}α_{μμ}| φ_{μ}^{1/q} = O(φ_{k}^{1/q}).

Remark 4. We are not yet able to provide necessary conditions. The first author has observed that the proof of the necessity-part of Theorem 4 in [4] in incorrect.

Absolute φ -summability factors with a power 9

5 Further sufficient conditions

THEOREM 4. Assume that φ satisfies (7) and that the method B satisfies the conditions (15),

∑_{k=0}^{n} |β̄_{nk}| = O(1),  (39)

∑_{n=k}^{∞} φ_{n}^{p−1} |β̄_{nk}| = O(φ_{k}^{p−1}),  (40)

∑_{n=k}^{∞} φ_{n}^{p−1} |b̄_{nk}| = O(β_{kk}φ_{k}^{p−1})  (41)

Then sufficient conditions for ε ∈ (φ−|A^{α}|_{p} , φ−|B|_{p}) are

∑_{n=0}^{∞} φ_{n}^{p−1} |G_{n}|^{p} |v_{n}|^{p} < ∞,  (42)

∑_{n=0}^{∞} φ_{n}^{p−1} |\frac{ε_{n}β_{nn}}{α_{nn}} | |v_{n}|^{p} < ∞,  (43)

∑_{k=0}^{n} |λ̄_{nk}| = O(1),  (44)

\frac{ε_{n+1}}{α_{n+1,n+1}} = O(\frac{ε_{n}}{α_{nn}}).  (45)

where the numbers λ̄_{nk} are defined by (28) and v_{n} is defined by (2) and satisfies (3).

Proof. As in the proof of Theorem 1 we have

w_{n} = ∑_{k=0}^{n} γ̄_{nk}v_{k}.

We must prove that (3) implies

∑ φ_{n}^{p−1} |w_{n}|^{p} < ∞.

By using the decomposition formula (27) and |u + v|^{p} ≤ 2^{p}(|u|^{p} + |v|^{p}) it suffices to show that

I_{1} := ∑_{n=0}^{∞} φ_{n}^{p−1} |∑_{k=0}^{n} β̄_{nk}G_{k}v_{k}|^{p} < ∞

and

I_{2} := ∑_{n=0}^{∞} φ_{n}^{p−1} |∑_{k=0}^{n} λ̄_{nk}v_{k}|^{p} < ∞.

By Hölder’s inequality, (39), (40), (42) we obtain

\begin{aligned}I_{1} & ≤ & ∑_{n=0}^{∞} φ_{n}^{p−1} (∑_{k=0}^{n} |β̄_{nk}| |G_{k}|^{p} |v_{k}|^{p}) (∑_{k=0}^{n} |β̄_{nk}|)^{p/q} \\ & = & O(1) ∑_{k=0}^{∞} |G_{k}|^{p} |v_{k}|^{p} ∑_{n=k}^{∞} φ_{n}^{p−1} |β̄_{nk}| = O(1) ∑_{k=0}^{∞} φ_{k}^{p−1} |G_{k}|^{p} |v_{k}|^{p} < ∞. \end{aligned}

Again by Hölder’s inequality and now by (44) we get

\begin{aligned}I_{2} & ≤ & ∑_{n=0}^{∞} φ_{n}^{p−1} (∑_{k=0}^{n} |λ̄_{nk}| |v_{k}|^{p}) (∑_{k=0}^{n} |λ̄_{nk}|)^{p/q} \\ & = & O(1) ∑_{k=0}^{∞} |v_{k}|^{p} ∑_{n=k}^{∞} φ_{n}^{p−1} |λ̄_{nk}| . \end{aligned}

By (28) and (41) we obtain

∑_{n=k}^{∞} φ_{n}^{p−1} |λ̄_{nk}| ≤ ∑_{n=k}^{k+α−1} φ_{n}^{p−1} ∑_{ν=k}^{n} |b̄_{nν}| ( ∑_{μ=ν+1}^{k+α} |η̄_{μk}ε_{μ}|)

+ ∑_{n=k+α}^{∞} φ_{n}^{p−1} ∑_{ν=k}^{k+α−1} |b̄_{nν}| ( ∑_{μ=ν+1}^{k+α} |η̄_{μk}ε_{μ}|)

= O(1) ∑_{ν=k}^{k+α} φ_{ν}^{p−1} |β_{νν}| ( ∑_{μ=ν+1}^{k+α} |η̄_{μk}ε_{μ}|) .

From this using (7), (15), (45), (43) we obtain

I_{2} = O(1) ∑_{k=0}^{∞} |v_{k}|^{p} ∑_{ν=k}^{k+α} φ_{ν}^{p−1} |β_{νν}| ∑_{μ=ν+1}^{k+α} |η̄_{μk}ε_{μ}|
= O(1) ∑_{k=0}^{∞} |v_{k}|^{p} φ_{k}^{p−1} |β_{kk}| ∑_{μ=k+1}^{k+α} |η̄_{μk}ε_{μ}|
= O(1) ∑_{k=0}^{∞} |v_{k}|^{p} φ_{k}^{p−1} |\frac{β_{kk}ε_{k}}{α_{kk}} | ∑_{μ=1}^{α} D̄_{μ} < ∞.

6 Applications to special methods

We now use Theorem 4 to obtain summability factors for special methods.

Let A = M_{a} and B = M_{b} with a_{n} ∼ n^{α}L_{1}(n) and b_{n} ∼ n^{β}L_{2}(n) , where α, β > −1 and the functions L_{1} and L_{2} are slowly varying (see [6], p.6 for the definition and relevant Absolute φ -summability factors with a power 11 properties). Let φ_{n} ∼ n^{γ}L_{3}(n), γ > 0 , and L_{3} be also a slowly varying function. As usual

A_{n} := ∑_{k=0}^{n} a_{k}, B_{n} := ∑_{k=0}^{n} b_{k}.

Then we have the following

COROLLARY 1. For the weighted means methods M_{a} and M_{b} the conditions

\frac{A_{n}b_{n}}{B_{n}a_{n}} ε_{n} = O(1),  (46)
ε_{n} = O(1),  (47)
\frac{A_{n}}{a_{n}} Δε_{n} = O(1)  (48)

are sufficient for ε ∈ (φ−|M_{a}|_{p} , φ−|M_{b}|_{p}) provided that the sequence φ is, as above, with γ(p − 1) < β + 1 .

Proof. The methods M_{a} are A^{1} methods (see [3], p.116). Furthermore (see [3], p.113)

β̄_{nk} = \{ \begin{matrix}\frac{b_{n}B_{k−1}}{B_{n}B_{n−1}} , & 1 ≤ k ≤ n, \\ 0, & otherwise, \end{matrix}

and (see [3], p.184, and [4], p.201)

G_{n} = \frac{A_{n}}{a_{n}} ε_{n} − \frac{A_{n−1}}{a_{n}} ε_{n+1}.

Since (B_{k−1}) is non-decreasing condition (39) follows. Furthermore B_{n} ∼ (β+1)^{−1}n^{β+1}L_{2}(n) (see [6], Proposition 1.5.8, p.26) and hence (15) is satisfied. The fulfilment of the conditions (40) and (41) is seen by applying Proposition 1.5.10 in [6], p.27 (or cf. [3], pp.80-81 in particular cases), which implies for γ(p − 1) < β + 1 the following formula

∑_{n=k}^{∞} n^{γ(p−1)−β−2} \frac{L_{3}^{p−1}(n)}{L_{2}(n − 1)} = O(1)k^{γ(p−1)−β−1} \frac{L_{3}^{p−1}(k)}{L_{2}(k − 1)}.

Hence for γ(p − 1) < β + 1 our Theorem 4 is applicable for M_{b}. From (48) and (47) we see that

|G_{n}| = |\frac{A_{n}}{a_{n}} Δε_{n}| + |ε_{n+1}| = O(1)

and by (3) we obtain (42). Furthermore (43) is directly implied by (46) and (3). Using (46), (47) and again the asymptotics of regularly varying functions it follows that

∑_{k=0}^{n} |λ̄_{nk}| ≤ ∑_{k=0}^{n} |∑_{ν=k}^{n} b̄_{nν} ( ∑_{μ=ν+1}^{k+α} ε_{μ}η̄_{μk})| = ∑_{k=0}^{n} |∑_{ν=k}^{n} \frac{−b_{n}b_{ν}}{B_{n}B_{n−1}} ∑_{μ=ν+1}^{k+1} ε_{μ}η̄_{μk}|

= \frac{b_{n}}{B_{n}B_{n−1}} ∑_{k=0}^{n} |b_{k}ε_{k+1}η̄_{k+1,k}| = \frac{b_{n}}{B_{n}B_{n−1}} ∑_{k=0}^{n} O(B_{k})

= O(1)\frac{nb_{n}}{B_{n}} = O(1),

since by (47), (48), (46) we obtain

\begin{aligned}ε_{k+1}η̄_{k+1,k} & = & −\frac{ε_{k+1}A_{k−1}}{a_{k}} = G_{k} − \frac{ε_{k}A_{k}}{a_{k}} \\ & = & O(1) + O(B_{k}/b_{k}) = O(B_{k}/b_{k}). \end{aligned}

By (47) and the special choice of weights (45) is obtained in the same way. By Theorem 4 the collorary is proved.

Remark 5.

(i) The above Corollary 1 contains certain cases of summability methods discussed in [21], especially Corollary 4, p.317.

(ii) One can also use Theorem 4 resp. Corollary 1 to obtain Theorem 1 in [7], p.146. There summability factors of type ε ∈ (φ−|M_{a}|_{p} , ψ−|M_{b}|_{p}) are considered. The condition (6) on p.146 in [7], however, allows to switch from the sequence φ to ψ and vice versa and allows Theorem 4 to be applied.

For a second application we consider the method B = Z_{β} of Silverman-Szasz (see [18], p.347, [3], p.100). We have b_{nk} = a_{n−k}/A_{n}, where a_{n} = 1 for n < β and a_{n} = 0 for n ≥ β. Hence A_{n} = n + 1 for n < β and A_{n} = β for n ≥ β. Therefore

1/β ≤ β_{kk} ≤ 1  (49)

and condition (9) is satisfied. Since b_{nk} = 0 for k ≤ n − β, then b̄_{nk} = 0 for k ≤ n − β and n ≥ k + β + 1 (see [3], p.50). Therefore the conditions (19) and (20) follow from (7). By (49) we obtain that (23) implies

G_{k} = ∑_{ν=0}^{α} |η̄_{ν+k,k}|O(α_{ν+k,ν+k}) = O(1) ∑_{ν=0}^{α} D̄_{ν} = O(1),

and consequently the conditions (21) and (22) follow from (23) and (7), because

β̄_{nk} = A_{n−k}Δ̄(1/A_{n}) + (1/A_{n−1})Δ̄A_{n−k}

(see [3], p.51 and p.96) and therefore β̄_{nk} = 0 for k ≤ n − β and n ≥ k + 1 + β. The necessity of (23) follows from a result of Jakimovski and Russell (see [10], Theorem 1 and formulas (5) and (17) with x_{k} = δ_{kν} ). Thus from Theorem 3 we obtain the following Absolute φ -summability factors with a power 13

COROLLARY 2. Assume that φ satisfies (7) and (14). Then ε ∈ (φ−|A^{α}|_{p} , φ−|Z_{β}|_{p}) if and only if

ε_{k} = O(α_{kk})

is satisfied.

In particular for the convergence method E = Z_{1} we have a related result (see [5], Theorem 2).

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Simson Baron and Rüdiger Kiesel

Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel, Department of Stochastics, University of Ulm, 89069 Ulm, Germany.