Absolute φ -convergence factors with a power

Simson Baron and Rüdiger Kiesel

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

We give necessary and sufficient conditions for a sequence (ε_{n}) of complex numbers to be convergence factors of type (φ−|A|_{p} , φ−|E|_{p}) , where A is a normal matrix of summability satisfying the Moore-Kangro condition and E is the convergence matrix.

1 Introduction and main result

Let φ = (φ_{n}) be a sequence of positive numbers. The series

∑ u_{n}  (1)

of complex numbers u_{n} is said to be φ -absolute convergent with the power p ≥ 1 or briefly, φ−|E|_{p} -convergent, if

∑ φ_{n}^{p−1} |u_{n}|^{p} < ∞.  (2)

Unless otherwise indicated the 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. Let

v_{n} = ∑_{k=0}^{n} ᾱ_{nk}u_{k}.  (3)

1AMS-Subject classification: 40 F

2 Absolute φ -convergence factors with a power

The series (1) is called φ -absolute A-summable with the power p ≥ 1 , or briefly φ−|A|_{p} -summable, if (see [14], p.314)

∑ φ_{n}^{p−1} |v_{n}|^{p} < ∞.  (4)

In particular, if A = E , the identity matrix, (4) is the same as (2).

Let ε = (ε_{n}) be a sequence of complex numbers. The numbers ε_{n} are called convergence factors of type (φ−|A|_{p} , φ−|E|_{p}) , if for any φ−|A|_{p} -summable series (1) the series

∑ ε_{n}u_{n}  (5)

is φ−|E|_{p} -convergent. The short notation is: ε ∈ (φ−|A|_{p} , φ−|E|_{p}) .

In this note we discover sufficient conditions such that ε ∈ (φ−|A|_{p} , φ−|E|_{p}) for p > 1 for a general class of matrices A, namely normal matrices satisfying the so-called Moore-Kangro condition, (see [3], p.190),

∑ nD_{n} < ∞,  (6)

where

D_{n} := sup_{k} |α_{n+k,n+k}η_{n+k,k}|

and (η_{nk}) := (α_{nk})^{−1} denotes the inverse series-to-sequence transform (which exits since A is normal).

To find such conditions one can use the theory of matrix transformations from l^{p} to l^{p} . For such transformations sufficient conditions are known ([11], p.174, Theorem 9; [12], pp. 55-56), namely: a triangular matrix G = (γ̄_{nk}) transforms any sequence (x_{n}) ∈ l^{p} in a sequence (y_{n}) ∈ l^{p} if

sup_{n≥0} ∑_{k=0}^{n} \left|{ \atop γ̄_{nk}}\right| < ∞  (7)

and

sup_{k≥0} ∑_{n=k}^{∞} \left|{ \atop γ̄_{nk}}\right| < ∞.  (8)

On the other hand necessary conditions are also known ([9], Theorem 1 and formulas (5),(17) with x_{k} = δ_{kν} ; and [7], p.467), namely

sup_{n≥0} ∑_{k=0}^{n} \left|{ \atop γ̄_{nk}}\right|^{q} < ∞  (9)

and

sup_{k≥0} ∑_{n=k}^{∞} \left|{ \atop γ̄_{nk}}\right|^{p} < ∞,  (10)

where q denotes always the conjugate exponent, i.e. 1/p + 1/q = 1 .

In the following we use the notations: m := [n/2] and

η_{n} := ∑_{k=0}^{n} η_{nk}.

We now formulate our main theorem.

THEOREM 1. Let A be a normal method of summability satisfying the condition (6) and let φ = (φ_{n}) be nondecreasing and satisfy the conditions

φ_{n} = O(φ_{k}),  m < k ≤ n,  (11)
\frac{φ_{n}^{1/q}}{n} = O (\frac{φ_{k}^{1/q}}{k} ) , 1 ≤ k ≤ n,  (12)

∑_{k=0}^{m} φ_{k}^{−1/q} = O(nφ_{n}^{−1/q}).  (13)

Then the numbers ε_{n} are convergence factors of type (φ−|A|_{p} , φ−|E|_{p}) for p > 1 , if the following conditions are fullfilled:

ε_{n} = O(α_{nn}),  (14)
nε_{n}η_{n} = O(1),  (15)
∑_{n=k}^{∞} φ_{n}^{1/q} |ε_{n}η_{n}| = O(φ_{k}^{1/q}).  (16)

Remark 1.(a) For the special sequence φ_{n} = n with n ≥ 1 and φ_{0} = 1 Theorem 1 is proved in [2].

(b) As choices for φ_{n} (satisfying (11)-(13)) sequences like n^{γ}L(n) with 0 < γ < q and L a slowly varying function are always possible, i.e. L is a positive measurable function, defined on some neighbourhood [X, ∞) of infinity, and satisfies

L(λx)/L(x) → 1 (x → ∞) ∀λ > 0

(see [5], p.6 for further information). Especially φ_{n} = (n + 1) log(n + 1) can always be used, since q > 1 .

Remark 2.(a) Riesz-Means M_{a} generated by a sequence a = (a_{n}) of nonnegative numbers, with A_{n} := a_{0} + … + a_{n} → ∞ as n → ∞ satisfy (6),(15),(16), since D_{n} = η_{n} = 0 ∀n ≥ 2 , (compare [3], p.58 and p.116). For such methods (14) reduces to

ε_{n} = O(a_{n}/A_{n}).

4 Absolute φ-convergence factors with a power

In case a_{n} = n^{γ}L(n) with γ > −1 and L slowly varying, we have A_{n} ∼ (γ + 1)^{−1}n^{γ+1}L(n) . Therefore the choice φ_{n} = A_{n}/a_{n} is possible and we obtain the so-called |M_{a}|_{p} -methods, (see [6], p.40).

(b) Cesàro-methods C_{α} satisfy (6), (15), (16) for α ≥ 0 (cf. [3], p.58 and p.192) and (14) is in this case

ε_{n} = O(n^{−α}).

(c) For Nörlund methods N_{b} generated by a sequence b = (b_{n}) with B_{n} := b_{0} + b_{1} + … + b_{n} satisfying B_{k} = O(B_{n}), k ≤ n, we have D_{n} = O(d_{n}) , where

∑ d_{n}x^{n} = 1/ ∑ B_{n}x^{n}

and the condition (6) reduces to ∑ n |d_{n}| < ∞ (cf. [13], p.44).

(d) Generally one can use A^{α} -methods as introduced in [1], p.108, that is methods A for which η_{nk} = 0 for k < n − α − 1 . This methods always satisfy (6) and so only (14),(15),(16) have to be verified. A large class of such methods is given by the generalized Nörlund methods (N, a, b) , (see [8, 10]), for which the series

∑ c_{n}x^{n} = 1/ ∑ a_{n}x^{n}

reduces to a polynomial (see the special cases above).

In the next section we start by giving a proof of Theorem 1. In the third section we discuss conditions on the matrix A so that (14) becomes a necessary and sufficient condition for ε ∈ (φ−|A|_{p} , φ−|E|_{p}) .

2 Proof of Theorem 1

Denote α := 1/p and β := 1/q, furthermore x_{n} := φ_{n}^{β}v_{n} and y_{n} := φ_{n}^{β}ε_{n}u_{n} . From (3) we obtain

y_{n} = ∑_{k=0}^{n} γ̄_{nk}x_{k},  (17)

where

γ̄_{nk} = (φ_{n}/φ_{k})^{β}ε_{n}η̄_{nk}.

Since |y_{n}|^{p} = φ_{n}^{p−1} |ε_{n}u_{n}|^{p} the series (5) is φ−|E|_{p} -convergent and therefore we have ε ∈ (φ−|A|_{p} , φ−|E|_{p}) if and only if the matrix transformation (17) preserves the space l^{p} .

Simson Baron and Rüdiger Kiesel 5

We begin by checking the condition (7). By formula (9.6) from [3] we obtain

γ̄_{nk} = (φ_{n}/φ_{k})^{β}ε_{n} (η_{n} − ∑_{ν=0}^{k−1} η_{nν}) , k ≥ 1,  (18)
γ̄_{n0} = (φ_{n}/φ_{0})^{β}ε_{n}η_{n}.

By (13), (11), (15) we obtain from this formula

∑_{k=0}^{n} \left|{ \atop γ̄_{nk}}\right| ≤ φ_{n}^{β} |ε_{n}η_{n}| ∑_{k=0}^{n} φ_{k}^{−β} + φ_{n}^{β}B_{n} = O(1) + φ_{n}^{β}B_{n}

where

B_{n} := |ε_{n}| ∑_{k=1}^{n} φ_{k}^{−β} ∑_{ν=0}^{k−1} |η_{nν}| .

By (14) we obtain

B_{n} = O(1) ∑_{k=0}^{n} φ_{k}^{−β} ∑_{ν=0}^{k} |α_{nn}η_{nν}|
= O(1) ∑_{k=0}^{n} φ_{k}^{−β} ∑_{ν=0}^{k} D_{n−ν}
= O(1) ∑_{k=0}^{n} φ_{k}^{−β} ∑_{ν=n−k}^{n} D_{ν} = O(1)(C_{n} + E_{n}),

where, since m = [n/2]

C_{n} := ∑_{k=0}^{m} φ_{k}^{−β} ∑_{ν=n−k}^{n} D_{ν}, E_{n} := ∑_{k=m}^{n} φ_{k}^{−β} ∑_{ν=n−k}^{n} D_{ν}.

By (6) and (13) we obtain for n ≥ 1

\begin{aligned}φ_{n}^{β}C_{n} & = & O(1/n)φ_{n}^{β} ∑_{k=1}^{m} φ_{k}^{−β} ∑_{ν=n−k}^{n} νD_{ν} \\ & = & O(1/n)φ_{n}^{β} ∑_{k=1}^{m} φ_{k}^{−β} = O(1) \end{aligned}

and (11) and (6) give

\begin{aligned}φ_{n}^{β}E_{n} & = & φ_{n}^{β} ∑_{k=m}^{n} φ_{k}^{−β} ∑_{ν=n−k}^{n} D_{ν} = O(1) ∑_{k=m}^{n} ∑_{ν=n−k}^{n} D_{ν} \\ & = & O(1) ∑_{ν=0}^{n} νD_{ν} = O(1). \end{aligned}

6 Absolute φ -convergence factors with a power Thus (7) follows from (6),(11),(13),(14),(15).

To establish condition (8) we have, in view of (18), to show that

∑_{n=k}^{∞} φ_{n}^{β} (|ε_{n}η_{n}| + |ε_{n} ∑_{ν=0}^{k−1} η_{nν} |) = O(φ_{k}^{β}), k ≥ 1.

By (16) it only remains to show that

F_{k} = O(φ_{k}^{β}), k ≥ 1,

where

F_{k} := ∑_{n=k}^{∞} φ_{n}^{β} |ε_{n} ∑_{ν=0}^{k−1} η_{nν} | .

Denote

ψ_{n} := φ_{n}^{β} − φ_{n−1}^{β},

with φ_{−1} := 0 . Since φ is nondecreasing, we have ψ_{n} ≥ 0 , and therefore

F_{k} ≤ ∑_{n=k}^{∞} ∑_{j=0}^{n} ψ_{j} ∑_{ν=0}^{k} |ε_{n}η_{nν}| = O(1)(G_{k} + H_{k}),

where

G_{k} := ∑_{ν=0}^{k} ∑_{j=0}^{k} ψ_{j} ∑_{n=k}^{∞} |ε_{n}η_{nν}| , H_{k} := ∑_{ν=0}^{k} ∑_{j=k}^{∞} ψ_{j} ∑_{n=j}^{∞} |ε_{n}η_{nν}| .

From (14) and (6) we get

G_{k} = ∑_{ν=0}^{k} ∑_{j=0}^{k} ψ_{j} ∑_{n=k−ν}^{∞} |ε_{n+ν}η_{n+ν,ν}| = O(1) ∑_{j=0}^{k} ψ_{j} ∑_{ν=0}^{k} ∑_{n=k−ν}^{∞} D_{n}
= O(1) ∑_{j=0}^{k} ψ_{j} (∑_{n=0}^{k} D_{n} ∑_{ν=k−n}^{k} 1 + ∑_{n=k+1}^{∞} D_{n} ∑_{ν=0}^{k} 1)
= O(1) ∑_{j=0}^{k} ψ_{j} ∑_{n=0}^{∞}(n + 1)D_{n} = O(1) ∑_{j=0}^{k} ψ_{j} = O(φ_{k}^{β}).

Finally by (14)

H_{k} = ∑_{j=k}^{∞} ψ_{j} ∑_{ν=0}^{k} ∑_{n=j−ν}^{∞} |ε_{n+ν}η_{n+ν,ν}| = O(1)(K_{k} + L_{k}),

where

\begin{aligned}K_{k} & := & ∑_{j=k}^{∞} ψ_{j} ∑_{n=j−k}^{j} D_{n} ∑_{ν=j−n}^{k} 1 = ∑_{j=k}^{∞} ψ_{j} ∑_{n=j−k}^{j} D_{n}(k + n − j + 1), \\ L_{k} & := & ∑_{j=k}^{∞} ψ_{j} ∑_{n=j+1}^{∞} D_{n} ∑_{ν=0}^{k} 1 ≤ ∑_{j=k}^{∞} ψ_{j} ∑_{n=j}^{∞}(k + 1)D_{n}. \end{aligned}

From (12) and (6) follows

K_{k} = ∑_{n=0}^{k} D_{n} ∑_{j=k}^{n+k}(n + k − j + 1)ψ_{j} + ∑_{n=k+1}^{∞} D_{n} ∑_{j=n}^{n+k}(n + k − j + 1)ψ_{j}

≤ ∑_{n=0}^{k}(n + 1)D_{n} ∑_{j=k}^{n+k} ψ_{j} + (k + 1) ∑_{n=k}^{∞} D_{n} ∑_{j=n}^{n+k} ψ_{j}

= ∑_{n=0}^{k}(n + 1)D_{n}(φ_{n+k}^{β} − φ_{k−1}^{β}) + (k + 1) ∑_{n=k}^{∞} D_{n}(φ_{n+k}^{β} − φ_{n−1}^{β})

= ∑_{n=0}^{k}(n + 1)(n + k)D_{n}O (\frac{φ_{k}^{β}}{k} ) + (k + 1) ∑_{n=k}^{∞}(n + k)D_{n}O (\frac{φ_{k}^{β}}{k} )

= O (φ_{k}^{β}) ∑_{n=0}^{k}(n + 1)D_{n} + O (φ_{k}^{β}) ∑_{n=0}^{k} 2nD_{n} = O(φ_{k}^{β}).

Also from (12) and (6) we obtain

\begin{aligned}L_{k} & = & (k + 1) ∑_{n=k}^{∞} D_{n} ∑_{j=k}^{n} ψ_{j} = (k + 1) ∑_{n=k}^{∞} D_{n} (φ_{n}^{β} − φ_{k−1}^{β}) \\ & = & O(1)(k + 1) ∑_{n=k}^{∞} nD_{n} \frac{φ_{k}^{β}}{k} = O(φ_{k}^{β}). \end{aligned}

Thus Theorem 1 is proved

3 Necessarity of the conditions

Theorem 2. Let φ be nondecreasing and satisfy the conditions (11),(12),(13) and the method of summability A satisfy (6) and α_{n0} = 1. Then the numbers ε_{n} are convergence factors of type (φ−|A|_{p} , φ−|E|_{p}) for p > 1 if and only if condition (14) is fullfilled.

Proof. If α_{n0} = 1 , then η_{n} = δ_{n0} (see [3], p.58, Theorem 9.2). Hence conditions (15) and (16) are automatically true. Furthermore by ([3], p.57) we have

γ̄_{nn} = ε_{n}η_{nn} = ε_{n}/α_{nn}

and we can use (9) or (10) to obtain the necessarity of (14). By Theorem 1 we get the sufficiency of (14).

Remark 3. With the exception of the general A^{α} methods all examples above obey the restriction α_{n0} = 1 .

Remark 4. For the case φ_{n} = n, n ≥ 1 and φ_{0} = 1 Theorem 2 is proved in [2].

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