Hindawi Publishing Corporation
Journal of Probability and Statistics
Volume 2016, Article ID 5862107, 10 pages
http://dx.doi.org/10.1155/2016/5862107
Research Article
Cesàro Summable Sequence Spaces over the
Non-Newtonian Complex Field
ULur Kadak
Department of Mathematics, Faculty of Sciences and Arts, Bozok University, Turkey
Correspondence should be addressed to Uğur Kadak; [email protected]
Received 12 September 2015; Accepted 27 October 2015
Academic Editor: Chin-Shang Li
Copyright © 2016 Uğur Kadak. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
𝑝
𝑝
can be considered the sets of all sequences that are strongly summable to zero, strongly summable,
The spaces 𝜔0 , 𝜔𝑝 , and 𝜔∞
and bounded, by the Cesàro method of order 1 with index 𝑝. Here we define the sets of sequences which are related to strong
Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the 𝛽-duals of
the new spaces and characterize matrix transformations on them into the sets of ∗-bounded, ∗-convergent, and ∗-null sequences
of non-Newtonian complex numbers.
1. Introduction
The theory of sequence spaces is the fundamental of summability. Summability is a wide field of mathematics, mainly in
analysis and functional analysis, and has many applications,
for instance, in numerical analysis to speed up the rate of convergence, in operator theory, the theory of orthogonal series,
and approximation theory. Also, the concepts of statistical
convergence have been studied by various mathematicians.
In recent years, generalizations of statistical convergence have
appeared in the study of strong integral summability and
the structure of ideals of bounded continuous functions on
locally compact spaces. Many important sequence spaces
arise in a natural way from different notions of summability,
that is, ordinary, absolute, and strong summability. The first
two cases may be considered as the domains of the matrices
that define the respective methods; the situation, however,
is different and more complicated in the case of strong
summability. Many authors have extensively developed the
theory of the matrix transformations between some sequence
spaces; we refer the reader to [1–6].
As an alternative to the classical calculus, Grossman and
Katz [7–9] introduced the non-Newtonian calculus consisting of the branches of geometric, quadratic, and harmonic
calculus, and so forth. All these calculi can be described
simultaneously within the framework of a general theory.
They decided to use the adjective non-Newtonian to indicate
any of calculi other than the classical calculus. Every property
in classical calculus has an analogue in non-Newtonian
calculus which is a methodology that allows one to have
a different look at problems which can be investigated via
calculus. In some cases, for example, for wage-rate (in dollars,
euro, etc.) related problems, the use of bigeometric calculus
which is a kind of non-Newtonian calculus is advocated
instead of a traditional Newtonian one.
Many authors have extensively developed the notion of
multiplicative calculus; see [10–12] for details. Also some
authors have also worked on the classical sequence spaces
and related topics by using non-Newtonian calculus [13–15].
Further Kadak [16] and Kadak et al. [17, 18] have matrix
transformations between certain sequence spaces over the
non-Newtonian complex field and have generalized RungeKutta method with respect to the non-Newtonian calculus.
The main focus of this work is to extend the strong Cesàro
summable sequence spaces defined earlier to their generalized sequence spaces over the non-Newtonian complex field
by using various generator functions, that is, exp and 𝑞𝑟
generators.
2. Preliminaries, Background, and Notations
Arithmetic is any system that satisfies the whole of the
ordered field axioms whose domain is a subset of R. There
2
Journal of Probability and Statistics
are infinitely many types of arithmetic, all of which are
isomorphic, that is, structurally equivalent.
A generator is a one-to-one function whose domain is R
and whose range is a subset R𝛼 of R where R𝛼 = {𝛼(𝑥) :
𝑥 ∈ R}. Each generator generates exactly one arithmetic,
and conversely each arithmetic is generated by exactly one
generator. If 𝐼(𝑥) = 𝑥 for all 𝑥 ∈ R, then 𝐼 is called identity
function whose inverse is itself. In the special cases 𝛼 = 𝐼 and
𝛼 = exp, 𝛼 generates the classical and geometric arithmetic,
respectively. By 𝛼-arithmetic, we mean the arithmetic whose
domain is R and whose operations are defined as follows. For
𝑥, 𝑦 ∈ R𝛼 and any generator 𝛼,
.
−1
−1
𝛼-addition 𝑥 + 𝑦 = 𝛼 {𝛼 (𝑥) + 𝛼 (𝑦)}
.
𝛼-subtraction 𝑥 − 𝑦 = 𝛼 {𝛼−1 (𝑥) − 𝛼−1 (𝑦)}
.
𝛼-multiplication 𝑥 × 𝑦 = 𝛼 {𝛼−1 (𝑥) × 𝛼−1 (𝑦)}
.
−1
(1)
−1
𝛼-division 𝑥 / 𝑦 = 𝛼 {𝛼 (𝑥) ÷ 𝛼 (𝑦)}
.
𝛼-order 𝑥 < 𝑦 ⇐⇒ 𝛼−1 (𝑥) < 𝛼−1 (𝑦) .
As an example if we choose exp function from R to the set
Rexp ⊆ R+ ,
𝛼 : R 󳨀→ Rexp
𝑥 󳨃󳨀→ 𝑦 = 𝛼 (𝑥) = 𝑒𝑥 ,
(2)
.
𝑛
𝑘=1
𝑘=1
−1
= 𝛼 {𝛼 (𝑥1 ) + ⋅ ⋅ ⋅ + 𝛼 (𝑥𝑛 )}
Definition 1 (see [13]). Let 𝑋 = (𝑋, 𝑑𝛼 ) be an 𝛼-metric space.
Then the basic notions can be defined as follows:
(a) A sequence 𝑥 = (𝑥𝑘 ) is a function from the set N into
the set R𝛼 . The 𝛼-real number 𝑥𝑘 denotes the value of
the function at 𝑘 ∈ N and is called the 𝑘th term of the
sequence.
(b) A sequence (𝑥𝑛 ) in 𝑋 = (𝑋, 𝑑𝛼 ) is said to be 𝛼. .
convergent if, for every given 𝜀 > 0 (𝜀 ∈ R𝛼 ), there
exist 𝑛0 = 𝑛0 (𝜀) ∈ N and 𝑥 ∈ 𝑋 such that
.
.
𝑑𝛼 (𝑥𝑛 , 𝑥) = |𝑥𝑛 − 𝑥|𝛼 < 𝜀 for all 𝑛 > 𝑛0 and is denoted
𝛼
by 𝛼 lim𝑛→∞ 𝑥𝑛 = 𝑥 or 𝑥𝑛 󳨀
→ 𝑥, as 𝑛 → ∞.
(c) A sequence (𝑥𝑛 ) in 𝑋 = (𝑋, 𝑑𝛼 ) is said to be 𝛼-Cauchy
. .
if for every 𝜀 > 0 there is 𝑛0 = 𝑛0 (𝜀) ∈ N such that
.
𝑑𝛼 (𝑥𝑛 , 𝑥𝑚 ) < 𝜀 for all 𝑚, 𝑛 > 𝑛0 .
2
.
.
𝑥3𝛼 = 𝑥2𝛼 × 𝑥 = 𝛼 {𝛼−1 {𝛼 [𝛼−1 (𝑥) × 𝛼−1 (𝑥)]} × 𝛼−1 (𝑥)}
(3)
3
= 𝛼 {[𝛼−1 (𝑥)] }
(6)
..
.
.
𝛼-division 𝑥 / 𝑦 = 𝑒{ln 𝑥/ ln 𝑦} = 𝑥1/ ln 𝑦 .
.
𝑝
𝑥𝑝𝛼 = 𝑥(𝑝−1)𝛼 × 𝑥 = 𝛼 {[𝛼−1 (𝑥)] } ,
Following Grosmann and Katz [8] we give the infinitely
many 𝑞𝑟 -arithmetic, of which the quadratic arithmetic and
harmonic arithmetic are special cases for 𝑟 = 2 and 𝑟 = −1,
respectively. The function 𝑞𝑟 : R → R𝑞 ⊆ R and its inverse
𝑞𝑟−1 (𝑥) are defined as follows:
1/𝑟
𝑥>0
{𝑥 ,
{
𝑞𝑟 (𝑥) = {0,
𝑥=0
{
1/𝑟
,
𝑥 < 0,
−
(−𝑥)
{
𝑟
𝑥>0
{𝑥 ,
{
=
(𝑥) {0,
𝑥 = 0,
{
𝑟
,
𝑥 < 0,
−
(−𝑥)
{
∀𝑥𝑘 ∈ R𝛼 .
𝑥2𝛼 = 𝑥 × 𝑥 = 𝛼 {𝛼−1 (𝑥) × 𝛼−1 (𝑥)} = 𝛼 {[𝛼−1 (𝑥)] }
𝛼-subtraction 𝑥 − 𝑦 = 𝑒{ln 𝑥−ln 𝑦} = 𝑥 ÷ 𝑦
𝑞𝑟−1
(5)
−1
.
𝛼-addition 𝑥 + 𝑦 = 𝑒{ln 𝑥+ln 𝑦} = 𝑥 ⋅ 𝑦
.
𝑛
∑ 𝑥𝑘 = 𝛼 { ∑ 𝛼−1 (𝑥𝑘 )}
𝛼
Throughout this paper, we define the 𝑝th 𝛼-exponent 𝑥𝑝𝛼
and 𝑞th 𝛼-root 𝑥(1/𝑞)𝛼 of 𝑥 ∈ R𝛼 by
and 𝛼-arithmetic turns out to be Geometric arithmetic:
𝛼-multiplication 𝑥 × 𝑦 = 𝑒{ln 𝑥 ln 𝑦} = 𝑥ln𝑦 = 𝑦ln𝑥
One can easily conclude that the 𝛼-summation can be
written as follows:
(4)
(𝑟 ∈ R \ {0}) .
If 𝑟 = 1 then the 𝑞𝑟 -calculus is reduced to the classical
calculus.
𝛼
and √𝑥
= 𝑥(1/2)𝛼 = 𝑦 provided there exists 𝑦 ∈ R𝛼 such that
2𝛼
𝑦 = 𝑥.
2.1. ∗-Arithmetic. Suppose that 𝛼 and 𝛽 be two arbitrarily selected generators and “star-” also be the ordered
pair of types of arithmetic (𝛽-arithmetic, 𝛼-arithmetic).
.
..
..
.. ..
.
.
.
The sets (R𝛽 , + , − , × , / ) and (R𝛼 , + , − , × , / ) are complete ordered fields and 𝑏𝑒𝑡𝑎(𝑎𝑙𝑝ℎ𝑎)-generator generates
𝑏𝑒𝑡𝑎(𝑎𝑙𝑝ℎ𝑎)-arithmetic, respectively. Definitions given for 𝛽arithmetic are also valid for 𝛼-arithmetic. Also 𝛼-arithmetic
is used for arguments and 𝛽-arithmetic is used for values; in
particular, changes in arguments and values are measured by
𝛼-differences and 𝛽-differences, respectively.
Definition 2 (see [15]). (a) The ∗-limit of a function 𝑓,
denoted by ∗ lim𝑥→𝑎 𝑓(𝑥) = 𝑏, at an element 𝑎 in R𝛼 is, if
it exists, the unique number 𝑏 in R𝛽 such that
Journal of Probability and Statistics
∗
3
.. ..
. .
.. 󵄨 ..
󵄨
lim 𝑓 (𝑥) = 𝑏 ⇐⇒ ∀𝜀 > 0 , ∃𝛿 > 0 ∋ 󵄨󵄨󵄨󵄨𝑓 (𝑥) − 𝑏󵄨󵄨󵄨󵄨𝛽 < 𝜀
𝑥→𝑎
A function 𝑓 is ∗-continuous at a point 𝑎 in R𝛼 if and only
if 𝑎 is an argument of 𝑓 and ∗ lim𝑥→𝑎 𝑓(𝑥) = 𝑓(𝑎). When
𝛼 and 𝛽 are the identity function 𝐼, the concepts of ∗-limit
and ∗-continuity are identical with those of classical limit and
classical continuity.
(b) The isomorphism from 𝛼-arithmetic to 𝛽-arithmetic
is the unique function 𝜄 (iota) which has the following three
properties:
(i) 𝜄 is one to one.
.
..
.
..
.
..
𝜄 (𝑢 + V) = 𝜄 (𝑢) + 𝜄 (V) ;
𝜄 (𝑢 × V) = 𝜄 (𝑢) × 𝜄 (V) ;
(8)
..
𝜄 (𝑢 / V) = 𝜄 (𝑢) / 𝜄 (V) ;
..
.
2.2. Non-Newtonian Complex Field and Some Inequalities.
.
.
.
.
..
.. .. ..
Let 𝑎 ∈ (R𝛼 , + , − , × , / ) and 𝑏 ∈ (R𝛽 , + , − , × , / ) be
arbitrarily chosen elements from corresponding arithmetic.
Then the ordered pair (𝑎, 𝑏) ∈ R𝛼 × R𝛽 ⊆ R2 is called a ∗point. The set of all ∗-points is called the set of ∗-complex
numbers and is denoted by C∗ ; that is,
∗
C fl {𝑧 = (𝑎, 𝑏) | 𝑎 ∈ R𝛼 , 𝑏 ∈ R𝛽 } .
(9)
Define the binary operations addition (⊕) and multiplication
(⊙) of ∗-complex numbers 𝑧1∗ = (𝑎1 , 𝑏1 ) and 𝑧2∗ = (𝑎2 , 𝑏2 ):
..
= (𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 )
= (𝛼 {𝛼−1 (𝑎1 ) + 𝛼−1 (𝑎2 )} , 𝛽 {𝛽−1 (𝑏1 ) + 𝛽−1 𝑏2 })
∗
⊙ : C × C 󳨀→ C
..
Then the pair (𝑋, 𝑑∗ ) and 𝑑∗ are called a non-Newtonian
metric (∗-metric) space and a ∗-metric on 𝑋, respectively.
The ∗-distance 𝑑∗ between two arbitrarily elements 𝑧1∗ =
(𝑎1 , 𝑏1 ) and 𝑧2∗ = (𝑎2 , 𝑏2 ) of the set C∗ is defined by
𝑑∗ (𝑧1∗ , 𝑧2∗ )
.
= √ [𝜄 (𝑎1 − 𝑎2 )]
𝛽
2𝛽 ..
..
2𝛽
+ (𝑏1 − 𝑏2 )
(11)
2
2
= 𝛽 {√[𝛼−1 (𝑎1 ) − 𝛼−1 (𝑎2 )] + [𝛽−1 (𝑏1 ) − 𝛽−1 (𝑏2 )] } .
Up to now, we know that C∗ is a field and the distance
between two points in C∗ is computed by the function 𝑑∗ .
Let 𝑧∗ = (𝑎, 𝑏) ∈ C∗ be an arbitrary element. The distance
function 𝑑∗ (𝑧∗ , 0∗ ) is called ∗-norm of 𝑧∗ and is denoted by
..
..
. ..
‖ ⋅ ‖ . In other words, let 0∗ = ( 0 , 0 ) ∈ C∗ ; then
..
2
2
‖ 𝑧∗ ‖ = 𝑑∗ (𝑧∗ , 0∗ ) = 𝛽 {√(𝛼−1 {𝑎}) + (𝛽−1 {𝑏}) } .
(𝑧1∗ , 𝑧2∗ ) 󳨃󳨀→
∗
..
(NM1) 𝑑∗ (𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦,
..
⊕ : C∗ × C∗ 󳨀→ C∗
∗
for all 𝑎1 , 𝑎2 ∈ R𝛼 and 𝑏1 , 𝑏2 ∈ R𝛽 .
..
It turns
out.. that 𝜄(𝑥) = 𝛽{𝛼 (𝑥)} for
every 𝑥 in R𝛼 and
.
.
that 𝜄( 𝑛 ) = 𝑛 for every 𝛼-integer 𝑛. Since, for example,
.
..
𝑢 + V = 𝜄−1 {𝜄(𝑢) + 𝜄(V)}, it should be clear that any statement
in 𝛼-arithmetic can readily be transformed into a statement
in 𝛽-arithmetic.
⊕
(10)
(NM3) 𝑑∗ (𝑥, 𝑦) ≤ 𝑑∗ (𝑥, 𝑧) + 𝑑∗ (𝑧, 𝑦).
−1
.
𝛽 {𝛼−1 (𝑎1 ) 𝛽−1 (𝑏2 ) + 𝛽−1 (𝑏1 ) 𝛼−1 (𝑎2 )})
(NM2) 𝑑∗ (𝑥, 𝑦) = 𝑑∗ (𝑦, 𝑥),
V ≠ 0 ; 𝑢 ≤ V ⇐⇒ 𝜄 (𝑢) ≤ 𝜄 (V) .
𝑧2∗
𝑧1∗ ⊙ 𝑧2∗ = (𝛼 {𝛼−1 (𝑎1 ) 𝛼−1 (𝑎2 ) − 𝛽−1 (𝑏1 ) 𝛽−1 (𝑏2 )} ,
Definition 4. Let 𝑋 be a nonempty set and let 𝑑∗ : 𝑋 × 𝑋 →
R𝛽 be a function such that, for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, the following
axioms hold:
𝜄 (𝑢 − V) = 𝜄 (𝑢) − 𝜄 (V) ;
𝑧1∗
(𝑧1∗ , 𝑧2∗ ) 󳨃󳨀→
Following Grossman and Katz [8] we can give the definition of ∗-distance and some applications with respect to the
∗-calculus.
(iii) For any numbers 𝑢, V ∈ R𝛼 ,
∗
(7)
Theorem 3 (see [15]). (C∗ , ⊕, ⊙) is a field.
(ii) 𝜄 is from R𝛼 onto R𝛽 .
.
󵄨 . 󵄨 .
∀𝜀 ∈ R𝛽 , 󵄨󵄨󵄨󵄨𝑥 − 𝑎󵄨󵄨󵄨󵄨𝛼 < 𝛿 for 𝑥, 𝛿 ∈ R𝛼 .
..
(12)
..
Moreover, for all 𝑧1∗ , 𝑧2∗ ∈ C∗ we have 𝑑∗ (𝑧1∗ , 𝑧2∗ ) = ‖ 𝑧1∗ ⊖𝑧2∗ ‖
..
..
where 𝑑∗ is the induced metric from ‖ ⋅ ‖ norm.
Theorem 5 (see [15]). (C∗ , 𝑑∗ ) is a complete metric space,
where 𝑑∗ is defined by (11).
4
Journal of Probability and Statistics
Corollary 6 (see [15]). C∗ is a Banach space with the ∗-norm
..
..
‖ ⋅ ‖ which is defined by (12).
Definition 7. (a) Given a sequence 𝑧𝑘∗ = (𝑎𝑘 , 𝑏𝑘 ) of ∗-complex
numbers, the formal notation
∞
(ii) Let 𝛼 and 𝛽 be the same generators. Then
𝑧1∗
..
⊙ 𝑧∗ = (𝑎1 , 𝑏1 ) ⊙ (𝑎1 , − 𝑏1 )
2
𝑘=0
𝑘=0
𝑘=0
𝐵∗ (𝑥0 ; 𝑟) = {𝑥 ∈ 𝑋 | 𝑑∗ (𝑥, 𝑥0 ) < 𝑟} ,
= (𝛼 ∑ 𝑎𝑘 , 𝛽 ∑ 𝑏𝑘 )
..
𝐵∗ [𝑥0 ; 𝑟] = {𝑥 ∈ 𝑋 | 𝑑∗ (𝑥, 𝑥0 ) ≤ 𝑟}
is called an infinite series with ∗-complex terms, or simply
complex ∗-series for all 𝑘 ∈ N. Also, for integers 𝑛 ∈ N,
the finite ∗-sums 𝑠𝑛∗ = ∗∑𝑛𝑘=0 𝑧𝑘∗ are called the partial sums of
complex ∗-series. If the sequence ∗-converges to a complex
number 𝑠∗ then we say that the series ∗-converges and write
∗
∗
𝑠∗ = ∗∑∞
𝑘=0 𝑧𝑘 . The number 𝑠 is then called the ∗-sum of this
series. If (𝑠𝑛 ) ∗-diverges, we say that the series ∗-diverges or
that it is ∗-divergent.
∗
(b) A ∗-series ∗∑∞
𝑘=0 𝑧𝑘 is said to ∗-converge absolutely if
..
..
∞
∗
∗∑𝑘=0 ‖ 𝑧𝑘 ‖ = ℓ for some number ℓ ∈ R𝛽 .
(c) Let {𝑓𝑛 (𝑥)} be a sequence of functions from 𝐴 ⊆ R to
C∗ for each 𝑛. We say that {𝑓𝑛 (𝑥)} is uniformly ∗-convergent
to 𝑓 on 𝐴 if and only if, for each 𝑥 ∈ 𝐴 and for an arbitrary
.. ..
𝜖 > 0 (𝜖 ∈ R𝛽 ), there exists an integer 𝑁 = 𝑁(𝜖, 𝑥) such that
..
𝑑∗ (𝑓𝑛 (𝑥), 𝑓(𝑥)) < 𝜖 whenever 𝑛 > 𝑁.
be uniformly ∗(d) The series ∗∑∞
𝑘=0 𝑓𝑘 (𝑥) is said to
.. ..
convergent to 𝑓(𝑥) on 𝐴 if, given any 𝜀 > 0 , there exists an
integer 𝑛0 (𝜀) such that
𝑛
..
𝑑∗ (∗ ∑ 𝑓𝑘 (𝑥) , 𝑓 (𝑥)) < 𝜀
whenever 𝑛 ≥ 𝑛0 (𝜀) .
𝑘=0
(14)
𝑛 ..
.. 𝑝𝛽
(1/𝑝)𝛽
𝑘=0
..
𝑛 ..
.. 𝑝𝛽
(1/𝑝)𝛽
≤ (𝛽 ∑ ‖ 𝑧𝑘∗ ‖ )
𝑘=0
..
𝑛 ..
.. 𝑝𝛽
+ (𝛽 ∑ ‖ 𝑡𝑘∗ ‖ )
(15)
(1/𝑝)𝛽
.
𝑘=0
Remark 9. Let 𝑧1∗ = (𝑎1 , 𝑏1 ), 𝑧2∗ = (𝑎2 , 𝑏2 ) ∈ C∗ . Then the
following statements hold:
(i) One has
𝑧1∗ ⊘ 𝑧2∗ =
(𝑎1 , 𝑏1 )
⊘
(𝑎2 , 𝑏2 )
= (𝛼 {
𝛽{
𝛼−1 (𝑎1 ) 𝛼−1 (𝑎2 ) + 𝛽−1 (𝑏1 ) 𝛽−1 (𝑏2 )
2
(𝛼−1 (𝑎2 )) + (𝛽−1 (𝑏2 ))
2
𝛽−1 (𝑏1 ) 𝛼−1 (𝑎2 ) − 𝛼−1 (𝑎1 ) 𝛽−1 (𝑏2 )
(𝛼−1
2
(𝑎2 )) +
(𝛽−1
2
(𝑏2 ))
(17)
},
(16)
(18)
are ∗-neighborhood (or ∗-open (closed) ball) of centre 𝑥0
and radius 𝑟, respectively.
We see that an ∗-open ball of radius 𝑟 is the set of all points
in 𝑋 whose beta-distance from the center of the ball is less
than 𝑟 and we say directly from the definition that every ∗neighborhood of 𝑥0 contains 𝑥0 ; in other words, 𝑥0 is a point
of each of its ∗-neighborhoods.
Definition 11. Let (𝑋, 𝑑∗ ) be a ∗-metric space. Then the
followings are valid:
(i) 𝐺 ⊂ 𝑋 is called ∗-open set if and only if every point of
𝐺 has a ∗-neighborhood contained in 𝐺. Also 𝐺 ⊂ 𝑋
is called ∗-closed set if and only if its complement is
∗-open.
(ii) The ∗-interior 𝐺𝑜 is the largest ∗-open set contained
in 𝐺 and the ∗-closure 𝐺 is the smallest ∗-closed set
contained in 𝐺.
Definition 12 (usual ∗-topology). Consider the set of ∗complex numbers with
..
Proposition 8 (see [15]). Let 𝑝 ≥ 1 and 𝑧𝑘∗ , 𝑡𝑘∗ ∈ C∗ for 𝑘 ∈
{0, 1, 2, 3, . . . , 𝑛}. Then
(𝛽 ∑ ‖ 𝑧𝑘∗ ⊕ 𝑡𝑘∗ ‖ )
.. 2𝛽
..
(13)
∞
..
..
Definition 10. Given a point 𝑥0 ∈ 𝑋. Then, for a positive 𝛽real number 𝑟,
∑ 𝑧𝑘∗ = 𝑧0∗ ⊕ 𝑧1∗ ⊕ 𝑧2∗ ⊕ ⋅ ⋅ ⋅ ⊕ 𝑧𝑘∗ ⊕ ⋅ ⋅ ⋅
∗
∞
2
= (𝛼 {(𝛼−1 (𝑎1 )) + (𝛽−1 (𝑏1 )) } , 0 ) = ‖ 𝑧∗ ‖ .
𝜏 = {𝑆 ⊆ C∗ | ∀𝑥 ∈ 𝑆, ∃𝑟 > 0 ∋ 𝐵∗ (𝑥0 ; 𝑟) ⊆ 𝑆}
∗
(19)
∗
for all 𝑥0 ∈ C and 𝑟 ∈ R𝛽 . Then (C , 𝜏) is a topological space
and is called ∗-usual topology on C∗ .
Definition 13. (i) A topological ∗-vector (linear) space 𝑋 is
a ∗-vector space (see [17]) over the topological field that
endowed with a topology such that ∗-vector addition and
scalar multiplication are ∗-continuous functions.
(ii) A topological ∗-vector space is called ∗-normable if
the topology of the space can be induced by a ∗-norm.
Definition 14. A sequence space 𝜆 with a ∗-linear topology is
called a ∗K-space provided each of the maps 𝑝𝑖 : 𝜆 → C∗
defined by 𝑝𝑖 (𝑥) = 𝑥𝑖 is ∗-continuous for all 𝑖 ∈ N. A ∗Kspace is called a ∗FK-space provided 𝜆 is a complete linear
∗-metric space (see [15]). An ∗FK-space whose topology is
∗-normable is called a ∗BK-space.
Definition 15 (𝑝-∗normed space). Let 𝑋 be a real or complex
..
..
∗-linear space and let ‖ ⋅ ‖ 𝑝 be a function from 𝑋 to the set
..
..
R+𝛽 and 𝑝 > 0. Then the pair (𝑋, ‖ ⋅ ‖ 𝑝 ) is called a 𝑝-∗normed
}) .
..
..
space and ‖ ⋅ ‖ 𝑝 is a 𝑝-∗norm for 𝑋, if the following axioms
are satisfied for all elements 𝑥, 𝑦 ∈ 𝑋 and for all scalars 𝜆:
Journal of Probability and Statistics
..
..
5
..
.
..
(N1) ‖ 𝑥 ‖ 𝑝 = 0 ⇔ 𝑥 = 𝜃∗ (𝜃∗ = ( 0 , 0 )),
..
..
..
.. 𝑝𝛽 .. ..
..
.. ..
(N2) ‖ 𝜆 ⊙ 𝑥 ‖ 𝑝 = ‖ 𝜆 ‖
modulus),
..
..
.. ..
..
..
..
× ‖ 𝑥 ‖ 𝑝 ( ‖ 𝜆 ‖ is complex
..
(N3) ‖ 𝑥 ⊕ 𝑦 ‖ 𝑝 ≤ ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝 .
∞
Definition 16. (i) A ∗-linear map or ∗-linear operator 𝑇
between real (or complex) ∗-linear spaces 𝑋, 𝑌 is a function
.
.
.
.
.
.
𝑇 : 𝑋 → 𝑌 such that 𝑇(𝜆 × 𝑥 + 𝜇 × 𝑦) = 𝜆 × 𝑇𝑥 + 𝜇 × 𝑇𝑦 for
∗
∗
∗
all 𝜆, 𝜇 ∈ R𝛼 and similarly 𝑇(𝜆 ⊙𝑥⊕𝜇 ⊙𝑦) = 𝜆 ⊙𝑇𝑥⊕𝜇∗ ⊙𝑇𝑦
for all 𝜆∗ , 𝜇∗ ∈ C∗ and 𝑥, 𝑦 ∈ 𝑋.
(ii) Let 𝑋 and 𝑌 be two ∗-normed linear spaces. A ∗linear map 𝑇 : 𝑋 → 𝑌 is ∗-bounded if there is a constant
..
..
.. ..
..
.. .. ..
𝑀 ≥ 0 such that ‖ 𝑇𝑥 ‖ 𝑌 ≤ 𝑀 × ‖ 𝑥 ‖ 𝑋 for all 𝑥 ∈ 𝑋. We
denote the set of all ∗-linear maps 𝑇 : 𝑋 → 𝑌 by L(𝑋, 𝑌)
and the set of all ∗-bounded linear maps by B(𝑋, 𝑌).
3. Non-Newtonian Infinite Matrices
Let 𝑤∗ denote the set of all ∗-complex sequences 𝑥 = (𝑥𝑘 )∞
𝑘=0 .
∗
, 𝑐∗ , 𝑐0∗ for the sets of all ∗-bounded, ∗As usual, we write ℓ∞
convergent, ∗-null sequences and
ℓ𝑝∗
∗
= {𝑧 =
(𝑧𝑘∗ )
∗
∞ ..
∈𝜔 :∗∑
𝑘=1
provided the series on the right hand side converge. On the
other hand the series on the right hand side of (22) converges
if and only if
‖ 𝑧𝑘∗
∑ (𝛼−1 (𝜀𝑖𝑘 ) 𝛼−1 (𝜇𝑘𝑗 ) − 𝛽−1 (𝛿𝑖𝑘 ) 𝛽−1 (𝜂𝑘𝑗 )) ,
𝑘=0
(23)
∞
−1
‖
< ∞}
(20)
∑ (𝛼 (𝜀𝑖𝑘 ) 𝛽 (𝜂𝑘𝑗 ) + 𝛽 (𝛿𝑖𝑘 ) 𝛼 (𝜇𝑘𝑗 ))
are convergent classically. However the series (22) may ∗diverge for some, or all, values of 𝑖, 𝑗; the product 𝐴 ⊙ 𝐵 may
not exist.
∗
) = (𝜀𝑛𝑘 , 𝛿𝑛𝑘 ) be an infinite
Let 𝜇1∗ , 𝜇2∗ ⊂ 𝑤∗ , and 𝐴 = (𝑎𝑛𝑘
matrix of ∗-complex numbers. Then we say that 𝐴 defines a
matrix mapping from 𝜇1∗ into 𝜇2∗ and denote it by writing 𝐴 :
𝜇1∗ → 𝜇2∗ , if for every sequence 𝑧 = (𝑧𝑘∗ ) ∈ 𝜇1∗ the sequence
𝐴 ⊙ 𝑧 = {(𝐴𝑧)𝑛 }, the 𝐴-transform of 𝑧, exists and is in 𝜇2∗ . In
this way, we transform the sequence 𝑧 = (𝑧𝑘∗ ) = (𝜇𝑘 , 𝜂𝑘 ) with
𝜇𝑘 ∈ R𝛼 and 𝜂𝑘 ∈ R𝛽 , into the sequence {(𝐴𝑧)𝑛 } by
∗
{(𝐴𝑧)𝑛 } = ∗ ∑𝑎𝑛𝑘
⊙ 𝑧𝑘∗
𝑘
∞
𝐴 ⊕ 𝐵 = (𝑎𝑖𝑗∗ ⊕ 𝑏𝑖𝑗∗ )
= (𝜖𝑖𝑗 , 𝛿𝑖𝑗 ) ⊕ (𝜇𝑖𝑗 , 𝜂𝑖𝑗 )
..
= (𝜖𝑖𝑗 + 𝜇𝑖𝑗 , 𝛿𝑖𝑗 + 𝜂𝑖𝑗 )
= (𝛼 {𝛼−1 (𝜖𝑖𝑗 ) + 𝛼−1 (𝜇𝑖𝑗 )} , 𝛽 {𝛽−1 (𝛿𝑖𝑗 ) + 𝛽−1 (𝜂𝑖𝑗 )}) ,
(21)
𝜆∗ ⊙ 𝐴 = (𝜆∗ ⊙ 𝑎𝑖𝑗∗ )
..
= ( 𝜆 , 𝜆 ) ⊙ (𝜖𝑖𝑗 , 𝛿𝑖𝑗 )
= (𝛼 {𝜆𝜖𝑖𝑗 − 𝜆𝛿𝑖𝑗 } , 𝛽 {𝜆𝜖𝑖𝑗 + 𝜆𝛿𝑖𝑗 }) ,
.
..
(𝜆∗ = ( 𝜆 , 𝜆 ) ∈ C∗ ) ,
where 𝜀𝑖𝑗 , 𝜇𝑖𝑗 ∈ R𝛼 and 𝛿𝑖𝑗 , 𝜂𝑖𝑗 ∈ R𝛽 for all 𝑖, 𝑗 ∈ N. Also, the
product (𝐴 ⊙ 𝐵)𝑖𝑗 of 𝐴 = (𝑎𝑖𝑗∗ ) and 𝐵 = (𝑏𝑖𝑗∗ ) can be interpreted
as
∞
∗
∗
= ∗ ∑ 𝑎𝑖𝑘
⊙ 𝑏𝑘𝑗
𝑘=0
∞
= (𝛼 { ∑ (𝛼−1 (𝜀𝑖𝑘 ) 𝛼−1 (𝜇𝑘𝑗 ) − 𝛽−1 (𝛿𝑖𝑘 ) 𝛽−1 (𝜂𝑘𝑗 ))} , (22)
𝑘=0
∞
𝛽 { ∑ (𝛼−1 (𝜀𝑖𝑘 ) 𝛽−1 (𝜂𝑘𝑗 ) + 𝛽−1 (𝛿𝑖𝑘 ) 𝛼−1 (𝜇𝑘𝑗 ))})
𝑘=0
−1
..
.. 𝑝
A non-Newtonian infinite matrix 𝐴 = (𝑎𝑖𝑗∗ ) of ∗-complex
numbers is defined by a function 𝐴 from the set N×N into C∗ .
The addition (⊕) and scalar multiplication (⊙) of the infinite
matrices 𝐴 = (𝑎𝑖𝑗∗ ) = (𝜖𝑖𝑗 , 𝛿𝑖𝑗 ) and 𝐵 = (𝑏𝑖𝑗∗ ) = (𝜇𝑖𝑗 , 𝜂𝑖𝑗 ) are
defined by
.
−1
𝑘=0
for 1 ≤ 𝑝 < ∞.
.
−1
= (𝛼 { ∑ (𝛼−1 (𝜀𝑛𝑘 ) 𝛼−1 (𝜇𝑘 ) − 𝛽−1 (𝛿𝑛𝑘 ) 𝛽−1 (𝜂𝑘 ))} ,
𝑘=0
(24)
∞
𝛽 { ∑ (𝛼−1 (𝜀𝑛𝑘 ) 𝛽−1 (𝜂𝑘 ) + 𝛽−1 (𝛿𝑛𝑘 ) 𝛼−1 (𝜇𝑘 ))})
𝑘=0
for all 𝑘, 𝑛 ∈ N. Thus, 𝐴 ∈ (𝜇1∗ : 𝜇2∗ ) if and only if the series on
the right side of (24) ∗-converges for each 𝑛 ∈ N and every
𝑧 ∈ 𝜇1∗ , and we have 𝐴 ⊙ 𝑧 = {(𝐴𝑧)𝑛 }𝑛∈N ∈ 𝜇2∗ for all 𝑧 ∈ 𝜇1∗ . A
sequence 𝑧 is said to be 𝐴-summable to 𝛾 if 𝐴⊙𝑧 ∗-converges
to 𝛾 ∈ C∗ which is called 𝐴-∗ lim of 𝑧.
The Cesàro transform of a sequence 𝑧 = (𝑧𝑘∗ ) ∈ 𝑤∗ is
given by 𝐶1∗ ⊙ 𝑧 = {(𝐶1∗ 𝑧)𝑛 }𝑛∈N . Now, following Example 17
we may state the Cesàro summability with respect to the nonNewtonian calculus which is analogous to the classical Cesàro
summable.
∗
) by
Example 17. Define the matrix 𝐶1∗ = (𝑐𝑛𝑘
∗
𝑐𝑛𝑘
(𝛼, 𝛽)
1
1 ∗
1
{
(
) = (𝛼 (
),𝛽(
)) , 0 ≤ 𝑘 ≤ 𝑛, (25)
{
{
{ 𝑛+1
𝑛+1
𝑛+1
={
{
{
{ ∗
𝑘 > 𝑛.
{0 = (𝛼 (0) , 𝛽 (0)) ,
If we choose the generator functions as 𝛼 = exp and 𝛽 = exp,
then the infinite matrix can be written as follows:
6
Journal of Probability and Statistics
(𝑒, 𝑒)
(1, 1)
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
(√𝑒, √𝑒)
⋅⋅⋅
⋅⋅⋅
(√𝑒, √𝑒)
(1, 1)
(
3
3
3
3
3
3
( (√𝑒,
√𝑒)
√𝑒)
√𝑒)
(√𝑒,
(√𝑒,
⋅⋅⋅
(1, 1)
(
(
∗
(
𝑐𝑛𝑘 (exp, exp) = (
..
..
..
..
.
.
.
d
.
(
(
( 𝑛+1 𝑛+1
√𝑒, 𝑛+1
√𝑒) ( 𝑛+1
√𝑒, 𝑛+1
√𝑒) ⋅ ⋅ ⋅ ( 𝑛+1
√𝑒, 𝑛+1
√𝑒)
( √𝑒, √𝑒) ( 𝑛+1
..
.
(
..
.
..
.
The 𝐶1∗ -transform of a sequence 𝑥 = (𝑥𝑘∗ ) = (𝑎𝑘 , 𝑏𝑘 ) is the
sequence 𝑦 = (𝑦𝑛∗ ) defined by
𝑥𝑘∗
. ⊘
𝑘=1 𝑛
𝑛
𝑎
= (𝛼 ∑ .𝑘 ⋅,
𝑘=1 𝑛
𝑏
∑ ..𝑘 :)
𝛽
𝑘=1 𝑛
(27)
where (⊘ is ∗-complex division), 𝑎𝑘 ∈ R𝛼 , and 𝑏𝑘 ∈ R𝛽 for all
𝑘 ∈ N. Taking 𝛼 = exp and 𝛽 = exp, we obtain 𝐶1∗ -transform
of 𝑥 = (𝑥𝑘∗ ) = (𝑎𝑘 , 𝑏𝑘 ) as follows:
𝑦𝑛∗ = (𝐶1∗ ⊙ 𝑥)𝑛
1 𝑛
1 𝑛
= (exp { ∑ ln (𝑎𝑘 )} , exp { ∑ ln (𝑏𝑘 )})
𝑛 𝑘=1
𝑛 𝑘=1
= ((𝑎1 ⋅ ⋅ ⋅ 𝑎𝑛 )
1/𝑛
, (𝑏1 ⋅ ⋅ ⋅ 𝑏𝑛 )
)
⋅⋅⋅ )
)
)
.. )
).
. )
)
)
⋅⋅⋅
(26)
d )
4. Matrix Transformations Related to
the Strong Cesàro ∗-Summability
..
𝑛
1 𝑛
1 𝑛
= (𝛼 { ∑ 𝛼−1 (𝑎𝑘 )} , 𝛽 { ∑ 𝛽−1 (𝑏𝑘 )}) ,
𝑛 𝑘=1
𝑛 𝑘=1
1/𝑛
..
.
⋅⋅⋅
Let 𝑒 and 𝑒(𝑛) (𝑛 = 0, 1, . . .) be the sequences with 𝑒𝑘 = 1∗ for
all 𝑘 ∈ N and 𝑒𝑛(𝑛) = 1∗ and 𝑒𝑘(𝑛) = 0∗ for 𝑘 ≠ 𝑛. Also, if 𝑋 is
𝑦𝑛∗ = (𝐶1∗ ⊙ 𝑥)𝑛 = ∗ ∑
𝑛
⋅⋅⋅
⋅⋅⋅
(28)
),
a ∗BK-space and 𝑎 ∈ 𝜔∗ then we write ‖𝑎‖‡ = sup{ ‖ ∗∑ 𝑎𝑘 ⊙
..
..
..
..
𝑥𝑘 ‖ : ‖ 𝑥 ‖ = 1}. Moreover, a ∗BK-space 𝑋 ⊃ 0 is said to
have ∗AK if every sequence 𝑥 = (𝑥𝑘 )∞
𝑘=1 ∈ 𝑋 has a unique
∞
(𝑘)
representation 𝑥 = ∗∑𝑘=1 𝑥𝑘 ⊙ 𝑒 . If 𝑋 is a subset of 𝑤∗ then
𝑋𝛽 = {𝑎 ∈ 𝜔∗ : 𝑎 ⊙ 𝑥 ∈ 𝑐𝑠∗ for all 𝑥 ∈ 𝑋} is called the 𝛽-dual
of 𝑋.
Throughout the text we use the notation [𝐶1∗ ]𝑝 for strong
Cesàro ∗-summability of order 1 and index 𝑝. Maddox [19]
introduced and studied the sets of sequences that are strongly
summable and bounded with index 𝑝, (1 ≤ 𝑝 < ∞) by
the Cesàro method of order one. By taking into account the
𝑝
𝑝
(𝛼, 𝛽) of sequences that are
sets 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽), and 𝜔∞
strongly ∗-summable to zero, ∗-summable, and ∗-bounded
of index 𝑝 ≥ 1 are defined by
(𝑎𝑛 , 𝑏𝑛 ∈ Rexp ) .
.. 𝑝𝛽
..
𝑛
{
‖ 𝑥𝑘 ‖
𝑝
𝜔0 (𝛼, 𝛽) = {𝑥𝑘 = (𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : 𝛽 lim 𝛽 ∑
..
𝑛→∞
𝑛
𝑘=1
{
.. }
fl 0 }
}
(𝑎𝑘 ∈ R𝛼 , 𝑏𝑘 ∈ R𝛽 )
..
2
2 𝑝/2
1 𝑛
= {(𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : lim 𝛽 { ∑ [(𝛼−1 {𝑎𝑘 }) + (𝛽−1 {𝑏𝑘 }) ] } = 0 } ,
𝑛→∞
𝑛 𝑘=1
.. 𝑝𝛽
..
𝑛
{
‖ 𝑥𝑘 ⊝ ℓ ‖
𝜔𝑝 (𝛼, 𝛽) = {𝑥𝑘 = (𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : 𝛽 lim 𝛽 ∑
..
𝑛→∞
𝑛
𝑘=1
{
}
..
fl 0 for some ℓ ∈ C∗ }
}
..
2
2 𝑝/2
1 𝑛
= {(𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : lim 𝛽 { ∑ [(𝛼−1 {𝑎𝑘 } − 𝛼−1 {ℓ1 }) + (𝛽−1 {𝑏𝑘 } − 𝛽−1 {ℓ2 }) ] } = 0 , ℓ = (ℓ1 , ℓ2 ) ∈ C∗ } ,
𝑛→∞
𝑛 𝑘=1
..
.. 𝑝𝛽
𝑛
{
‖ 𝑥𝑘 ‖
𝑝
𝜔∞
(𝛼, 𝛽) = {𝑥𝑘 = (𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : sup 𝛽 ∑
..
𝑛
𝑛∈N 𝑘=1
{
}
..
..
2
2 𝑝/2
1 𝑛
: < ∞} = {(𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : sup 𝛽 { ∑ [(𝛼−1 {𝑎𝑘 }) + (𝛽−1 {𝑏𝑘 }) ] } < ∞} .
𝑛 𝑘=0
𝑛∈N
}
(29)
Journal of Probability and Statistics
7
..
.. 𝑝𝛽
𝑛
{
‖ 𝑥𝑘 ‖
{
{
:,
sup
∑
{
..
{
𝛽
{
𝑛
.. ..
{ 𝑛∈N 𝑘=1
‖𝑥‖† = {
(1/𝑝)𝛽
.. 𝑝𝛽
..
{
𝑛
{
‖ 𝑥𝑘 ‖
{
{
{
sup
(
,
:)
∑
..
{
𝛽
𝑛
𝑛∈N
𝑘=1
{
For instance,
𝑝
(i) 𝜔0 (exp, 𝑞𝑟 ) = {(𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : lim𝑛→∞ {(1/
𝑛) ∑𝑛𝑘=1 (ln2 (𝑎𝑘 ) + (𝑏𝑘 )2𝑟 )𝑝/2 }1/𝑟 = 0}, (𝑎𝑘 ∈ Rexp , 𝑏𝑘 ∈
R𝑞 , 𝑟 ∈ R+ , 𝑝 ≥ 1);
(ii) 𝜔𝑝 (𝑞𝑟 , exp) = {(𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ : lim𝑛→∞ {(1/
𝑛) ∑𝑛𝑘=1 [((𝑎𝑘 )𝑟 − (ℓ1 )𝑟 )2 + ln2 (𝑏𝑘 /𝑙2 )]𝑝/2 } = 0}, (𝑎𝑘 , ℓ1 ∈
R𝑞 ; 𝑏𝑘 , ℓ2 ∈ Rexp ).
..
(32)
(1/𝑝)𝛽
.. 𝑝𝛽
..
,
.
:)
Proof. The proof is straightforward (see [20]).
𝑝
First we give the 𝛽-duals of the spaces 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽),
𝑝
(𝛼, 𝛽). We prefer the notation ∑] and max] for the
and 𝜔∞
sum and maximum taken over all indices 𝑘 with 2] ≤ 𝑘 ≤
2]+1 − 1 and put
Proof. The proof is a routine verification and hence omitted.
𝑝
𝑝
Corollary 19. The sets 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽), and 𝜔∞
(𝛼, 𝛽)
(𝛼,𝛽)
are Banach spaces with the induced metric 𝑑𝑝
from the
(𝛼,𝛽)
M𝑝
..
corresponding norm ‖ 𝑥 ‖ † defined by
..
= {𝑧 ∈ 𝜔∗ : ‖𝑧‖M𝑝 < ∞} ,
(33)
where
∞
𝑝
:)
‖ 𝑥𝑘 ‖
= sup (𝛽 ∑
..
𝑛
𝑛∈N
𝑘=1
∞
‖ 𝑥𝑘 ⊝ 𝑦𝑘 ‖
{
{
{
:,
0 < 𝑝 < 1,
sup 𝛽 ∑
..
{
{
𝑛
(30)
{
{ 𝑛∈N 𝑘=1
fl {
(1/𝑝)𝛽
.. 𝑝𝛽
..
{
{
{ 𝑛 ‖ 𝑥𝑘 ⊝ 𝑦𝑘 ‖
}
{
{
{sup {𝛽 ∑
, 1 ≤ 𝑝 < ∞.
:}
..
{
𝑛
𝑛∈N
{
{ 𝑘=1
}
‖𝑧‖M(𝛼,𝛽)
𝑛
‖𝑥‖†𝜔𝑝 (𝛼,𝛽)
.. 𝑝𝛽
(1/𝑝)𝛽
.. 𝑝𝛽
‖𝑥 ‖
= sup (𝛽 ∑ 𝑘]𝛽
2
]∈N
]
𝑝
‖𝑥‖𝜔∞
(𝛼,𝛽)
𝑑𝑝(𝛼,𝛽) (𝑥, 𝑦)
..
1 ≤ 𝑝 < ∞.
𝑝
𝑝
..
(31)
Proposition 20. Let 1 ≤ 𝑝 < ∞. The sets 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽),
𝑝
(𝛼, 𝛽) are ∗BK-spaces with the (equivalent) ∗-norms
and 𝜔∞
𝑝
(𝛼, 𝛽) are
Theorem 18. The sets 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽), and 𝜔∞
(𝛼,𝛽)
∗
complete ∗-metric spaces over the field C with the metric 𝑑𝑝
defined by
𝑛
0 < 𝑝 < 1,
..
..
.. ]𝛽 ..
{
{
∑ 2 × max ‖ 𝑧𝑘 ‖ ,
{
𝛽
{
]
{
{ ]=0
{
={
{
(1/𝑞)𝛽
∞
{
.. 𝑞𝛽
..
{
.. (]/𝑝)𝛽 ..
{
{
×
(
‖
𝑧
‖
)
,
∑
∑
{𝛽 2
𝑘
𝛽
]
{ ]=0
𝑝 = 1,
(𝛼,𝛽)
(1 < 𝑝 < ∞; 𝑞 =
𝑝
),
(𝑝 − 1)
∀𝑧 ∈ M𝑝
.
(34)
In particular, we have
∞
‖𝑧‖M(exp,exp)
𝑝
..
for all 𝑧𝑘 = (𝑎𝑘 , 𝑏𝑘 ) ∈ 𝜔∗ where 𝑎𝑘 , 𝑏𝑘 ∈ Rexp = (0, ∞).
∗
Now, with regard to notation, for any matrix 𝐴 = (𝑎𝑛𝑘
) we
write, for 1/𝑝 + 1/𝑞 = 1, 𝑝 ≥ 1,
.. 𝑞𝛽
..
∗
𝐴𝑝] (𝑛) = (𝛽 ∑ ‖ 𝑎𝑛𝑘
‖ )
(1/𝑞)𝛽
(36)
]
and the case 𝑝 = 1 of (36) is interpreted as
..
..
∗
𝐴1] (𝑛) = max ‖ 𝑎𝑛𝑘
‖.
]
..
∞
..
..
2]
{
exp { ∑ 2] ln (max ‖ 𝑧𝑘 ‖ )} = ∏ (max ‖ 𝑧𝑘 ‖ ) , 𝑝 = 1,
{
{
]
]
{
{
]=0
]=0
={
1/𝑞
∞
{
.. 𝑞
..
{
{
{exp { ∑ 2]/𝑝 (∑ (ln ‖ 𝑧𝑘 ‖ ) ) } ,
1 < 𝑝 < ∞,
]
{
]=0
(37)
(35)
Remark 21. Let 𝑌 be a ∗-normed space and let 𝑇 be a ∗-linear
operator on the 𝑝-∗normed space 𝑋 into 𝑌. Then it is easy to
see that 𝑇 is ∗-continuous on 𝑋 if and only if there is a positive
..
..
..
.. ..
.. (1/𝑝)𝛽
on 𝑋.
constant 𝑀 ∈ R𝛽 such that ‖ 𝑇𝑥 ‖ 𝑌 ≤ 𝑀 × ‖ 𝑥 ‖ 𝑋
The ∗-norm on the left is the norm in 𝑌 and that on the
right is 𝑝-∗norm in 𝑋. It is also seen that Banach-Steinhaus
theorem in the non-Newtonian sense holds in a 𝑝-∗normed
space defined in Definition 15.
Also one can immediately conclude that [𝐶1∗ ]𝑝 is ∗complete in the metric generated by the ∗-norm. For
simplicity we denote [𝐶1∗ ]𝑝 by Ω, so that, in the case 𝑝 ≥ 1, Ω
8
Journal of Probability and Statistics
is a Banach space and, in the case 0 < 𝑝 < 1, Ω is a complete
𝑝-∗normed space. The dual space of Ω, that is, the space of
continuous ∗-linear functionals on Ω, will be denoted by Ω∗ .
To prove uniformly ∗-convergent, by using the conjugate
numbers 𝑝 and 𝑞 for 0 < 𝑝 < 1, then, for any positive integer
𝑛0 and for any 𝑚 ≥ 2𝑛0 , we obtain
∞ ..
Corollary 22 (cf. [19] (Banach-Steinhaus)). If (𝑇𝑛 ) is
a sequence of continuous ∗-linear operators on a 𝑝∗normed space 𝑋 into a ∗-normed space 𝑌 such that
..
..
..
𝛽
lim sup𝑛 ‖ 𝑇𝑛 𝑥 ‖ 𝑌 < ∞ on a second category set in 𝑋, then
..
sup𝑛 ‖𝑇𝑛 ‖ < ∞. It is important to remark that ‖𝑇𝑛 ‖ denotes the
..
..
.. ..
𝑘=𝑚
The next theorem characterizes all infinite matrices 𝐴
which map the space [𝐶1∗ ]𝑝 into the space 𝑐∗ of convergent
sequences of ∗-complex numbers.
Theorem 23. (𝐴𝑧)𝑛 is defined for each 𝑛 and the sequence
{(𝐴𝑧)𝑛 } is ∗-convergent, whenever 𝑧 ∈ Ω, if and only if,
(a) for 0 < 𝑝 < 1,
.. (]/𝑝)𝛽 ..
≤ 𝛽 ∑ 𝐴1]
]=𝑛0
(ii) 𝑀 = sup𝑛 {𝛽 ∑∞
]=0 2
× 𝐴1] (𝑛)} < ∞;
(i) as in (a) above,
.. (]/𝑝)𝛽 ..
∗
∗
(iii) 𝑑
=
∗
𝛽
∑
𝑘=1
∗
‖ 𝑎𝑛𝑘
→
where 𝑧𝑘
..
.. ..
∞ .. (]/𝑝)𝛽 ..
× max] ‖ 𝜉𝑘 ‖ ≤ 𝑀,
𝛽 ∑]=0 2
holds
..
..
∞
..
..
..
(43)
]
= 0 , (𝑛 → ∞).
for each 𝑛 and each ] ≥ 0. They are ∗-linear and ∗-continuous
since
.. ..
..
.. .. (]/𝑝)𝛽 .. ..
∗
‖ ∗ ∑𝑎𝑛𝑘
⊙ 𝑧𝑘 ‖ ≤ 𝐴1] (𝑛) × 2
.. 1/𝑝
× ‖𝑧‖𝑝 .
]
.. ..
..
.. ..
.. (1/𝑝)𝛽
.. .. ..
.. 𝑝𝛽
(38)
(1/𝑝)𝛽
.. (]/𝑝)𝛽 ..
≤ 𝛽 ∑ 𝐴1] (𝑛) × 2
..
.. (−1/𝑝)𝛽
× sgn 𝑧𝑛,𝑁] , 𝑧𝑘 = 0∗ (𝑘 ≠ 𝑁] ), for 0 ≤ ] ≤ 𝑠,
..
..
..
..
where 𝑁] is such that ‖ 𝑧𝑛,𝑁] ‖ = max] ‖ 𝑎𝑛,𝑘 ‖ . By (45) we
.. .. (]/𝑝)𝛽 ..
.. 1/𝑝
× ‖𝑧‖𝑝
..
..
≤𝑀×
..
.. 1/𝑝 ..
‖𝑧‖𝑝 <
∞.
∞
]=0
Again for 0 < 𝑝 < 1 we will show that the conditions (i) and
(ii) imply (iii). In fact we will prove that (ii) which implies
..
..
∗
∑∞
𝑘=0 ‖ 𝑎𝑛𝑘 ‖ is uniformly ∗-convergent (see Definition 7) and
this together with (i) gives
lim
𝑛
𝑧𝑁] = 2
have 𝛽∑𝑠]=0 𝐴1] (𝑛) × 2
]
∗
(45)
.. (]/𝑝)𝛽 ..
]
..
,
where ‖𝐴 𝑛 ‖ = sup𝑛 ‖ (𝐴𝑧)𝑛 ‖ × ‖ 𝑧 ‖ 𝑝
. Taking any
∗
integer 𝑠 > 0 and define 𝑧 ∈ Ω by 𝑧𝑘 = 0 for 𝑘 ≥ 2𝑠+1 and
..
..
(44)
∗
From Corollary 22, it follows that lim] ∗∑] 𝑎𝑛𝑘
⊙ 𝑧𝑘 = (𝐴𝑧)𝑛 is
∗
in Ω , whence
..
..
imply
(42)
‖ (𝐴𝑧)𝑛 ‖ ≤ ‖ 𝐴 𝑛 ‖ × ‖ 𝑧 ‖ 𝑝
∗
≤ 𝛽 ∑ (𝛽 ∑ ‖ 𝑎𝑛𝑘
⊙ 𝑧𝑘 ‖ )
]=0
(i)-(ii)
∗
⊙ 𝑧𝑘 󳨀→ ∗ ∑ 𝜉𝑘 ⊙ 𝑧𝑘
∑ 𝑎𝑛𝑘
⊙ 𝑧𝑘 ‖
∞
and
∗
‖ ∗ ∑𝑎𝑛𝑘
⊙ 𝑧𝑘 ‖ ∈ Ω∗ ,
∗
⊙ 𝑧𝑘 ‖
= 𝛽 ∑ 𝛽 ∑ ‖ 𝑎𝑛𝑘
]=0
ℓ,
(41)
and we deduce that
⊙
𝑧
is
absolutely
∗-convergent
and the last sum in (41)
𝜉
∑
𝑘
∗
..𝑘
goes to 0 as 𝑛 → ∞. From (39),
..
∞
.
∗
∗
⊖ 𝜉𝑘 )] ⊕ ∗ ∑ (𝑎𝑛𝑘
⊖ 𝜉𝑘 ) (𝑧𝑘 ⊖ ℓ)
∑ 𝜉𝑘 ⊙ 𝑧𝑘 ⊕ [ℓ ⊙ ∗ ∑ (𝑎𝑛𝑘
..
Proof. The proofs of necessity condition of (i) and (iii) and
the sufficiency of the conditions are routine verification and
hence omitted.
Firstly we prove the sufficiency when 0 < 𝑝 < 1, leaving
the necessities (ii) and (ii)󸀠 to the next. Now, consider that
the condition (ii) holds. Then the series defining (𝐴𝑧)𝑛 is
absolutely ∗-convergent (see Definition 7) for each 𝑛 ∈ N.
By using the known inequality we have
∞ ..
≤𝑀× 2
(40)
∗
⊙ 𝑧𝑘
∑ 𝑎𝑛𝑘
× 𝐴𝑝] (𝑛) < ∞,
∗
(∗∑∞
𝑘=0 𝑎𝑛𝑘 , 𝜉)
(𝑛) × 2
.. .. (𝑛0 /𝑞)𝛽
for every 𝑧 ∈ 𝜔∗ . Also, the sufficiency for the case 𝑝 ≥ 1 can
be obtained in a similar way.
Conversely suppose that (𝐴𝑧)𝑛 exists for each 𝑛 ≥ 1
whenever 𝑥 ∈ Ω. Thus, the functionals
(b) for 𝑝 ≥ 1,
(ii)󸀠 𝑀 = 𝛽 ∑∞
]=0 2
.. .. ]𝛽 ..
Thus,
∗
..
]
∞
..
..
∗
(i) 𝑑∗ (𝑎𝑛𝑘
, 𝜉𝑘 ) = 0 (𝑛 → ∞, 𝑘 fixed),
..
..
]=𝑛0
.. (−1/𝑝)𝛽
norm of 𝑇𝑛 , that is, ‖𝑇𝑛 ‖ = sup𝑛 ‖ 𝑇𝑛 𝑥 ‖ 𝑌 × ‖ 𝑥 ‖ 𝑋
, the
supremum being taken over all non-𝛽-zero elements of 𝑋.
∞
.. ..
∗
∗
‖ ≤ 𝛽 ∑ 𝛽 ∑ ‖ 𝑎𝑛𝑘
‖
∑ ‖ 𝑎𝑛𝑘
𝛽
∗
∑ 𝑎𝑛𝑘
∗
= ∗ ∑ 𝜉𝑘 .
(39)
≤ ‖𝐴 𝑛 ‖ for each 𝑛 and implies
󵄩 󵄩 ..
≤ 󵄩󵄩󵄩𝐴 𝑛 󵄩󵄩󵄩 < ∞.
.. .. (]/𝑝)𝛽 ..
∑ 𝐴1] (𝑛) × 2
𝛽
]=0
(46)
In fact the series (𝐴𝑧)𝑛 was absolutely ∗-convergent which
gives
∞
.. .. (]/𝑝)𝛽
󵄩󵄩 󵄩󵄩 ..
1
.
󵄩󵄩𝐴 𝑛 󵄩󵄩 ≤ 𝛽 ∑ 𝐴 ] (𝑛) × 2
]=0
(47)
Journal of Probability and Statistics
9
𝑝
From (46) and (47), it can easily be seen that
∞
.. .. (]/𝑝)𝛽
󵄩󵄩 󵄩󵄩
1
.
󵄩󵄩𝐴 𝑛 󵄩󵄩 = 𝛽 ∑ 𝐴 ] (𝑛) × 2
(48)
]=0
This leads us together with Corollary 22 to the fact that
∞
.. (]/𝑝)𝛽 .. 1
..
󵄩 󵄩
sup 󵄩󵄩󵄩𝐴 𝑛 󵄩󵄩󵄩 = sup {𝛽 ∑ 2
× 𝐴 ] (𝑛)} < ∞.
𝑛
𝑛
(49)
]=0
Similarly, one can prove that (ii)󸀠 is necessary condition. This
completes the proof.
Now, as an immediate consequence of Theorem 23, the
𝑝
following corollary shows the 𝛽-duals for 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽),
𝑝
and 𝜔∞ (𝛼, 𝛽).
𝑝
Corollary 24. The 𝛽-duals of the spaces 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽),
(𝛼,𝛽)
𝑝
and 𝜔∞
(𝛼, 𝛽) are M𝑝 .
Remark 25. The symbol “𝛽” used for 𝛽-dual has different
meaning from that of 𝛽 generator used in the text.
We may give with quoting the following proposition and
lemma without proofs (see [21, 22]) which are needed in the
proof of next theorem.
Proposition 26. Let 𝑋 be a ∗BK-space. Then we have 𝐴 ∈
∗
∗
) if and only 𝐴 𝑛 ∈ 𝑋𝛽 for all 𝑛 ∈ N where 𝐴 𝑛 = (𝑎𝑛𝑘
)
(𝑋 : ℓ∞
for the sequence in the 𝑛th row of 𝐴 for all 𝑘 ∈ N.
∗
) be an innite matrix. If
Lemma 27. Let 𝐴 = (𝑎𝑛𝑘
..
..
𝛽
‖𝐴 𝑛 ‖M(𝛼,𝛽) < ∞ and lim𝑛→∞ ‖𝐴 𝑛 ‖M(𝛼,𝛽) = 0 , then ‖𝐴 𝑛 ‖M(𝛼,𝛽)
𝑝
𝑝
𝑝
uniformly ∗-converges in 𝑛 ∈ N.
𝑝
∗
𝑝
∗
∗
(i) 𝐴 = (𝑎𝑛𝑘
) ∈ (𝜔∞
(𝛼, 𝛽) : ℓ∞
) = (𝜔0 (𝛼, 𝛽) : ℓ∞
) =
𝑝
∗
(𝜔 (𝛼, 𝛽) : ℓ∞ ) if and only if
󵄩 󵄩
sup 󵄩󵄩󵄩𝐴 𝑛 󵄩󵄩󵄩M(𝛼,𝛽) < ∞.
𝑝
..
𝑛∈N
(50)
∗
𝑝
) ∈ (𝜔∞
(𝛼, 𝛽) : 𝑐0∗ ) if and only if
(ii) 𝐴 = (𝑎𝑛𝑘
..
󵄩 󵄩
lim 󵄩󵄩𝐴 󵄩󵄩 (𝛼,𝛽) = 0 .
𝑛→∞ 󵄩 𝑛 󵄩M𝑝
(51)
𝑝
∗
) ∈ (𝜔0 (𝛼, 𝛽) : 𝑐0∗ ) if and only if (50) holds
(iii) 𝐴 = (𝑎𝑛𝑘
and
∗
lim 𝑎∗
𝑛→∞ 𝑛𝑘
= 0∗
(52)
for all 𝑘 ∈ N.
∗
(𝑎𝑛𝑘
)
(iv) 𝐴 =
hold and
𝑝
∈ (𝜔 (𝛼, 𝛽) :
∗
𝑐0∗ )
if and only if (50), (52)
∞
∗
lim ∗ ∑ 𝑎𝑛𝑘
= 0∗ .
𝑛→∞
𝑘=1
5. Concluding Remarks
Non-Newtonian calculus is a methodology that allows one to
have a different look at problems which can be investigated
via calculus. It should be clear that the non-Newtonian
calculus is a self-contained system independent of any other
system of calculus. Therefore the reader may be surprised
to learn that there is a uniform relationship between the
corresponding operators of this calculus and the classical
calculus.
In this paper we have introduced the sequence spaces
𝑝
𝑝
(𝛼, 𝛽) as a generalization of the
𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽), and 𝜔∞
𝑝
𝑝
𝑝
spaces 𝜔0 , 𝜔 , and 𝜔∞ of Maddox [19]. Our main purpose is
𝑝
to determine the 𝛽-duals of the new spaces 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽),
𝑝
and 𝜔∞ (𝛼, 𝛽) and is to characterize the classes of matrix
transformations from these spaces to any one of the spaces
∗
, 𝑐∗ , and 𝑐0∗ . As a future work we will try to obtain the
ℓ∞
characterizations of the classes of infinite matrices from the
𝑝
𝑝
(𝛼, 𝛽) to a sequence space
spaces 𝜔0 (𝛼, 𝛽), 𝜔𝑝 (𝛼, 𝛽), and 𝜔∞
∗
,
𝜆 over the non-Newtonian complex field different from ℓ∞
∗
∗
𝑐 , and 𝑐0 .
Competing Interests
Theorem 28. The following statements hold:
𝛽
∗
) follows
Proof. (i) Condition (50) for 𝐴 ∈ (𝜔0 (𝛼, 𝛽) : ℓ∞
from Proposition 26 and Corollary 24. Then the other parts
𝑝
𝑝
(𝛼, 𝛽) by
follows from the fact that 𝜔0 (𝛼, 𝛽) ⊂ 𝜔𝑝 (𝛼, 𝛽) ⊂ 𝜔∞
Proposition 20.
(ii) This condition is proved in the same way as in
Theorem 23 with 𝜉𝑘 = 0∗ for all 𝑘 ∈ N.
𝑝
(iii) Since 𝜔0 (𝛼, 𝛽) is a ∗BK-space with ∗AK by Propo∗ ∗
∗
the conditions
sition 20 and 𝑐 , 𝑐0 are closed subset of ℓ∞
𝑝
∗
follow from the characterization of (𝜔0 (𝛼, 𝛽) : ℓ∞
).
(iv) The conditions follow from those in (iii).
(53)
The author declares that there are no competing interests.
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