Math. Slovaca 61 (2011), No. 2, 245–256
DOUBLE SEQUENCE SPACES DEFINED BY A MODULUS
Ekrem Savas¸* — Richard F. Patterson**
(Communicated by J´an Bors´ık )
ABSTRACT. This paper begins with new definitions for double sequence spaces.
These new definitions are constructed, in general, by combining modulus function and nonnegative four-dimensional matrix. We use these definitions to establish inclusion theorems between various sequence spaces such as: IfA = (am,n,k,l) be a nonnegative four-dimensional matrix such that
supm,n
∞,∞
k,l=0,0
am,n,k,l< ∞
and letf be a modulus, then ω(A, f) ⊂ ω∞(A, f) and ω0(A, f) ⊂ ω∞(A, f).
2011c Mathematical Institute Slovak Academy of Sciences
1. Introduction and background
The class of sequences which are strongly Ces`aro summable with respect to a modulus was introduced by Maddox [4] as an extension of the definition of strongly Ces`aro summable sequences. Connor [2] further extended this notion to strong A-summability with respect to a modulus where A = (an,k) is a non- negative regular matrix. Using this definition Connor established connections between strong A-summability, strong A-summability with respect to a modu- lus, and A-statistical convergence. In 1900 Pringsheim presented a definition for convergence of double sequences in [6]. Following Pringshiem work, Hamilton and Robison in [3] and [7], respectively, presented a series of necessary and/or sufficient conditions on the entries of A = (am,n,k,l) that ensure the preserva- tion of Pringsheim type convergence on the following transformation of double
2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 42B15; Secondary 40C05.
K e y w o r d s: double sequences, P-convergent, modulus functions.
sequences (Ax)m,n=
∞,∞
k,l=0,0
am,n,k,lxk,l. Throughout this paper the four dimen- sional matrices and double sequences are of real-valued entries unless specified otherwise. The goals of this paper include the extension of some of the funda- mental theorems of ordinary (single) summability theory to multi-dimensional summability theory. These extension begins with the presentation of the follow- ing sequence spaces:
x ∈ s : P- lim
m,n
∞,∞
k,l=0,0am,n,k,lf(|xk,l|) = 0
,
x ∈ s : P- lim
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) = 0, for some L
,
x ∈ s : sup
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|) < ∞
,
where f is a modulus function and A is a nonnegative four dimensional matrix.
2. Definitions and preliminaries
Let s denote the set of all double sequences of complex numbers. By con- vergence of a double sequence we shall mean the convergence in the Pringsheim sense, that is, a double sequence x = [xk,l] has Pringsheim limit L (denoted by P- lim x = L) provided that given ε > 0 there exists N ∈ N such that
|xk,l− L| < ε whenever k, l > N ([6]). We shall also describe such an x more briefly as P-convergent. We shall denote the space of all P-convergent sequences by c2.
Ruckle and Maddox in [8] and [4], respectively, presented the following defi- nition:
2.1 A function f : [0,∞) → [0, ∞) is called a modulus provided that
(1) f (x) = 0 if and only if x = 0,
(2) f (x + y)≤ f(x) + f(y) for all x ≥ 0 and y ≥ 0, (3) f is increasing, and
(4) f is continuous from the right of 0.
Throughout this paper we shall examine our sequence spaces using the fol- lowing transformation:
2.2 Let A = (am,n,k,l) denote a four dimensional summability method that maps the complex double sequences x into the double sequence Ax where the mnth term to Ax is as follows:
(Ax)m,n=
∞,∞
k,l=1,1
am,n,k,lxk,l.
2.3 Let f be a modulus and A = (am,n,k,l) be a nonnegative four dimensional matrix of real entries with sup
m,n
∞,∞
k,l=0,0
am,n,k,l < ∞, then
ω0(A, f ) =
x ∈ s : P- lim
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|) = 0
,
ω(A, f ) =
x ∈ s : P- lim
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) = 0, for some L
,
ω∞(A, f ) =
x ∈ s : sup
m,n
∞,∞
k,l=0,0am,n,k,lf(|xk,l|) < ∞
.
If f (x) = x then the sequence spaces reduce to the following:
ω0(A) =
x ∈ s: P- lim
m,n
∞,∞
k,l=0,0am,n,k,l|xk,l| = 0
,
ω(A) =
x ∈ s: P- lim
m,n
∞,∞
k,l=0,0
am,n,k,l|xk,l− L| = 0, for some L
,
ω∞ (A) =
x ∈ s: sup
m,n
∞,∞
k,l=0,0
am,n,k,l|xk,l|
< ∞.
By specifying A and f the spaces in Definition 2.3 reduce to some well-known sequence spaces. For example, if A = (C, 1, 1) the sequence spaces become ω(f ), ω0(f ), and ω∞(f ) which are as follows:
ω0(f ) =
x ∈ s : P- lim
m,n mn1
m−1,n−1
k,l=0,0 f(|xk,l|) = 0
,
ω(f ) =
x ∈ s : P- lim
m,n mn1
m−1,n−1
k,l=0,0 f(|xk,l− L|) = 0, for some L
,
ω∞ (f ) =
x ∈ s : sup
m,n mn1
m−1,n−1
k,l=0,0 f(|xk,l|) < ∞
.
As a final illustration, if we take A = (C, 1, 1) and f (x) = x, we obtain the following spaces:
ω0 =
x ∈ s: P- lim
m,n mn1
m−1,n−1
k,l=0,0 |xk,l| = 0
,
ω =
x ∈ s: P- lim
m,n mn1
m−1,n−1
k,l=0,0 |xk,l− L| = 0, for some L
,
ω∞ =
x ∈ s: sup
m,n mn1
m−1,n−1
k,l=0,0 |xk,l| < ∞
.
3. Main results
In this section we shall establish the main properties of the sequence spaces in Definition 2.3.
3.1 ω0(A, f ), ω(A, f ), and ω∞(A, f ) are linear spaces over the complex field C.
The proof is obvious, and as of such, it is omitted.
3.2 If A = (am,n,k,l) be a nonnegative matrix such that supm,n
∞,∞
k,l=0,0
am,n,k,l< ∞
and let f be a modulus then ω(A, f )⊂ ω∞ (A, f ) and ω0(A, f )⊂ ω∞ (A, f ).
P r o o f. We shall establish the first inclusion, the second inclusion is clear. Let x ∈ ω(A, f ). Then by conditions (2) and (3) of the modulus function we are granted the following:
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|) ≤
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) + f(|L|)
∞,∞
k,l=0,0
am,n,k,l.
Also observe that there exists an integer Ml such that|L| ≤ Ml. Thus, we have
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|) ≤
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) + Mlf(1)
∞,∞
k,l=0,0
am,n,k,l.
Since sup
m,n
∞,∞
k,l=0,0
am,n,k,l < ∞ and x ∈ ω(A, f ), we are granted that x ∈
ω∞ (A, f ) and this completes the proof.
3.3 If A = (am,n,k,l) be a nonnegative matrix such that supm,n
∞,∞
k,l=0,0
am,n,k,l< ∞
and let f be a modulus then ω(A) ⊂ ω(A, f ), ω0(A) ⊂ ω0(A, f ), ω∞ (A) ⊂ ω∞ (A, f ).
P r o o f. The first two inclusions are easily proved. Thus, we will only establish the last inclusion. Let x∈ ω∞ (A), such that sup
m,n
∞,∞
k,l=0,0
am,n,k,l|xk,l| < ∞. Let ε > 0 and choose δ with 0 < δ < 1 such that f(t) < ε for 0 ≤ t ≤ δ. Let us consider the following equality
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|)
=
∞,∞
k,l=0,0
|xk,l|≤δ
am,n,k,lf(|xk,l|) +
∞,∞
k,l=0,0
|xk,l|>δ
am,n,k,lf(|xk,l|).
The properties of f grant us the following:
∞,∞
k,l=0,0
|xk,l|≤δ
am,n,k,lf(|xk,l|) ≤ ε
∞,∞
k,l=0,0
am,n,k,l. (3.1)
For|xk,l| > δ and the fact that
|xk,l| < |xk,l| δ <
1 +|xk,l| δ
where [t] denoted the integer part of t and from conditions (2) and (3) of the modulus function we have
f(|xk,l|) <
1 +
|xk,l| δ
f(1) ≤ 2f(1)|xk,l| δ .
Thus ∞,∞
k,l=0,0
|xk,l|>δ
am,n,k,lf(|xk,l|) ≤ 2f (1) δ
∞,∞
k,l=0,0
am,n,k,l|xk,l|.
The last inequality and (3.1) yields the following:
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|) ≤ ε
∞,∞
k,l=0,0
am,n,k,l+2f (1) δ
∞,∞
k,l=0,0
am,n,k,l|xk,l|.
Since sup
m,n
∞,∞
k,l=0,0
am,n,k,l< ∞ and x ∈ ω∞(A) we are granted that x∈ ω∞(A, f ).
This completes the proof.
3.4 If A = (am,n,k,l) be a nonnegative matrix such that supm,n
∞,∞
k,l=0,0
am,n,k,l< ∞
and let f be a modulus and β = lim
t→∞
f(t)t > 0 then ω(A) = ω(A, f ).
P r o o f. In the previous theorem we have shown that ω(A) ⊆ ω(A, f ). Now let β > 0. By definition of β we have f (t)≥ βt for all t ≥ 0. Since β > 0 we have t≤ 1βf(t) for all t ≥ 0. Note x ∈ ω(A, f ) implies
∞,∞
k,l=0,0
am,n,k,l|xk,l− L| ≤ 1 β
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|)
whence x∈ ω(A). This completes the proof.
3.5 If A = (am,n,k,l) has only positive entries and B = (bm,n,k,l) be a nonnegative matrix such that
bm,n,k,l
am,n,k,l
is bounded then ω∞ (A, f )⊂ ω∞(B, f ).
P r o o f. The proof is obvious thus omitted.
3.6 If A = (am,n,k,l) be a nonnegative matrix such that supm,n
∞,∞
k,l=0,0
am,n,k,l< ∞
and let f be a modulus then ω0(A, f ) and ω(A, f ) are complete linear topological spaces with the paranorm
g(x) = sup
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l|).
P r o o f. From the statements above the result for ω0(A, f ) is clear. Let us consider ω(A, f ). From Theorem 3.2 for each x∈ ω(A, f ), g(x) exists. Clearly g(θ) = 0, g(−x) = g(x), and g(x + y) ≤ g(x) + g(y). We now show that the scalar multiplication is continuous. First observe the following:
g(λx) = sup
m,n
∞,∞
k,l=0,0
am,n,k,lf(|λxk,l|) ≤ (1 + [|λ|])g(x),
where [|λ|] denotes the integer part of |λ|. In addition observe that x and λ → 0 implies g(λx)→ 0. For fixed λ, if x approaches 0 then g(λx) approaches 0. We
need to show that for fixed x, λ approaches 0 implies g(λx) approaches 0. Let x ∈ ω(A, f ) this implies that
P- lim
m,n
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) = 0.
Let ε > 0 and choose N such that
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) < ε
4 (3.2)
for m, n > N . Also, for each (m, n) with 1≤ m, n ≤ N, since
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) < ∞,
there exists an integer Mm,n such that
k,l>Mm,n
am,n,k,lf(|xk,l− L|) < ε 4. Let
M = max
1≤(m,n)≤N{Mm,n}.
We have for each (m, n) with 1≤ (m, n) ≤ N
k,l>M
am,n,k,lf(|xk,l− L|) < ε 4. Also from (3.2), for m, n > N we have
k,l>M
am,n,k,lf(|xk,l− L|) < ε 4. Thus M is an integer independent of (m, n) such that
k,l>M
am,n,k,lf(|xk,l− L|) < ε
4. (3.3)
Further for|λ| < 1 and for all (m, n)
∞,∞
k,l=0,0
am,n,k,lf(|λxk,l|)
=
∞,∞
k,l=0,0
am,n,k,lf(|λxk,l− λL + λL|)
≤
∞,∞
k,l=0,0
am,n,k,lf(|λxk,l− λL|) +
∞,∞
k,l=0,0
am,n,k,lf(|λL|)
≤
k,l>M
am,n,k,lf(|λxk,l− λL|) +
k,l≤M
am,n,k,lf(|λxk,l− λL|)
+
k≥Ml<M
am,n,k,lf(|λxk,l− λL|) +
k<M l≥M
am,n,k,lf(|λxk,l− λL|)
+ f (|λL|)
∞,∞
k,l=0,0
am,n,k,l.
(3.4)
For each (m, n) and by the continuity of f as λ→ 0 we have the following:
k,l≤M
am,n,k,lf(|λxk,l− λL|) + f(|λL|)
∞,∞
k,l=0,0
am,n,k,l → 0
in the Pringsheim sense. Now choose δ < 1 such that|λ| < δ implies
k,l≤M
am,n,k,lf(|λxk,l− λL|) + f(|λL|)
∞,∞
k,l=0,0
am,n,k,l< ε
4. (3.5) In the same manner we are granted
k≥Ml<M
am,n,k,lf(|λxk,l− λL|) < ε
4, (3.6)
and
k<Ml≥M
am,n,k,lf(|λxk,l− λL|) < ε
4. (3.7)
It follows from (3.3) through (3.7) that
∞,∞
k,l=0,0
am,n,k,lf(|λxk,l|) < ε for all (m, n).
Thus g(λx) approaches 0 as λ approaches 0. Therefore ω0(A, f ) is a paranormed linear topological space. Now let us show that ω0(A, f ) is complete with respect
to its paranorm topologies. Let (xsk.l) be a Cauchy sequence in ω0(A, f ). Then, we write g(xs− xt)→ 0 as s, t → ∞, to mean, as s, t → ∞ for all (m, n),
∞,∞
k,l=0,0
am,n,k,lf(|xsk,l− xtk,l|) → 0. (3.8)
Thus for each fixed k and l as s, t → ∞, since A = (am,n,k,l) is nonnegative, we are granted f (|xsk,l− xtk,l|) → 0 and by continuity of f, (xsk.l) is a Cauchy sequence in C for each fixed k and l. Since C is complete as s → ∞ we have xsk,l → xk,l for each (k, l). Now from (3.8), we have for ε > 0, there exists a natural number N such that
∞,∞
k,l=0,0 s,t>N
am,n,k,lf(|xsk,l− xtk,l|) < ε (3.9)
for all (m, n). Since for any fixed natural number M we have form (3.9)
∞,∞
k,l≤M s,t>N
am,n,k,lf(|xsk,l− xtk,l|) < ε
for all (m, n), by letting t→ ∞ in the above expression we obtain
∞,∞
k,l≤M s>N
am,n,k,lf(|xsk,l− xk,l|) < ε.
Since M is arbitrary, by letting M→ ∞ we obtain
∞,∞
k,l=0,0
am,n,k,lf(|xsk,l− xk,l|) < ε
for all (m, n). Thus g(xs− x) → 0 as s → ∞. Also (xs) being a sequence in ω(A, f ), by definition of ω(A, f ), for each s there exists Ls with
∞,∞
k,l=0,0
am,n,k,lf(|xsk,l− Ls|) → 0
as (m, n)→ ∞, whence, from the fact that sup
m,n
∞,∞
k,l=0,0
am,n,k,l< ∞ form the def- inition of modulus function condition (2), we have f (|Ls− Lt|) → 0 as s, t → ∞ and so Ls converges to L. Thus
∞,∞
k,l=0,0
am,n,k,lf(|xk,l− L|) → 0
as (m, n)→ ∞, thus x ∈ ω(A, f ) and this completes the proof.
Recall that a double sequence is called bounded if there exists a positive number M such that|xj,k| < M for all j and k. The notion of regularity for two dimensional matrix transformations was presented by Silverman and Toeplitz in [9] and [10], respectively. Following Silverman and Toeplitz, Robison and Hamilton presented the following four dimensional analog of regularity for double sequences in which they both added an additional assumption of boundedness.
This assumption was made because a double sequence which is P-convergent is not necessarily bounded.
3.1 The four dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.
In addition to this definition, Robison and Hamilton also presented the fol- lowing Silverman-Toeplitz type multidimensional characterization of regularity in [3] and [7]:
3.7 The four dimensional matrix A is RH-regular if and only if RH1: P- lim
m,nam,n,k,l= 0 for each k and l;
RH2: P- lim
m,n
∞,∞
k,l=1,1
am,n,k,l= 1;
RH3: P- lim
m,n
∞
k=1|am,n,k,l| = 0 for each l;
RH4: P- lim
m,n
∞
l=1|am,n,k,l| = 0 for each k;
RH5:
∞,∞
k,l=1,1|am,n,k,l| is P-convergent; and
RH6: there exist positive numbers A and B such that
k,l>B|am,n,k,l| < A.
Finally we conclude this paper by presenting the multidimensional analogue of Connor ([2, Theorems 6, 8]).
3.8 Let x∈ s be bounded, f be a modulus function and A a non- negative RH-regular summability matrix method. Then the double sequence x is strongly A-summable to L with respect to the modulus function f if and only if x is strongly A-summable to L, that is, ω(A, f )∩ l∞ = ω(A)∩ l∞ .
Before presenting the next theorem, let us consider the following notions.
Let K ⊂ N × N, be a two dimensional set of positive integers. Then the A = (am,n,k,l)-density of K is given by
δA2(K) = P- lim
m,n
(k,l)∈K
am,n,k,l
provided that the limit exists. The notion of double asymptotic density for double sequence was presented by Mursaleen and Edely in [5].
3.2 A double complex number sequence x is said to be A-statis- tically P-convergent to L if, for every positive ε,
δA2
{(k, l) : |xk,l− L| ≥ ε}
= 0.
3.9 Let f be a modulus function and A a nonnegative RH-regular summability matrix method.
(1) If x∈ s is strongly A-summable to L with respect to f , then x is A-stat- istically convergent to L.
(2) If x∈ s is bounded and A-statistically convergent to L, then x is strongly A-summable to L with respect to the modulus function f.
The proof of these theorems are omitted since they can be proved by using the techniques present by Connor in [2].
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Received 4. 12. 2008 Accepted 21. 1. 2010
* Istanbul Commerce University Department of Mathematics Uskudar/Istanbul
TURKEY
E-mail : ekremsavas@yahoo.com
** Department of Mathematics and Statistics University of North Florida
Building 11
Jacksonville, Florida, 32224 USA
E-mail : rpatters@unf.edu
