Recent Developments on Sequence Spaces and Compact Operators with ApplicationsView this Special Issue
Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field
In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the field and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets over to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.
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. This subsection serves as a motivation of what follows. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. The idea is to assign a limit of some sort to divergent sequences or series by considering a transform of a sequence or series rather than the original sequence or series. One can ask why we employ the special transformations represented by infinite matrices instead of general linear operators. The answer to this question is that, in many cases, the most general linear operators between two sequence spaces are given by an infinite matrix. Many authors have extensively developed the theory of the matrix transformations between some sequence spaces we refer the reader to [1–13].
As an alternative to the classical calculus, Grossman and Katz [14–16] 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. We 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.
Bashirov et al. [17, 18] have recently concentrated on the non-Newtonian calculus and gave the results with applications corresponding to the well-known properties of derivatives and integrals in the classical calculus. Also, Uzer  has extended the non-Newtonian calculus to the complex valued functions and was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Further, Misirli and Gurefe have introduced multiplicative Adams Bashforth-Moulton methods for differential equations in . Quite recently, Kadak [21, 22] have determinated Köthe-Toeplitz dual between classical sets of sequences over the non-Newtonian complex field and have constructed Hilbert spaces over the non-Newtonian field.
The main purpose of the present paper is to characterize some matrix classes between certain sequence spaces over the non-Newtonian complex field.
2. -Arithmetics and Some Related Applications
A generator is a one-to-one function whose domain is and whose range is a subset of , the set of real numbers. Each generator generates exactly one arithmetic, and, conversely, each arithmetic is generated by exactly one generator. For example, the identity function generates classical arithmetic and exponential function generates geometric arithmetic. As a generator, we choose the function such that its basic algebraic basic algebraic operations are defined as follows: for all where the non-Newtonian real field as in .
The -positive real numbers, denoted by , are the numbers in such that ; the -negative real numbers, denoted by , are those for which . The -zero, , and the -one, , turn out to be and . Further, holds for all . Thus the set of all -integers turns out to be the following:
Definition 1 (see ). Let be a nonempty set and let be a function such that for all , , , then the following axioms hold: (NM1) if and only if ,(NM2),(NM3). Then, the pair and are called a non-Newtonian metric space and a non-Newtonian metric on , respectively.
Definition 2 (see ). Let be a non-Newtonian metric space. Then the basic notions can be defined as follows. (a)A sequence is a function from the set into the set . The -real number denotes the value of the function at and is called the kth term of the sequence.(b)A sequence in a metric space is said to be -convergent if for every given there exists an and such that for all and is denoted by or , as .(c)A sequence in is said to be non-Newtonian Cauchy (-Cauchy) if for every there is an such that for all .
Remark 3 (see ). Let . Then the number is called the -square and is denoted by . Let . Then is called the -square root of and is denoted by . Further, for each we can write for all .
The -absolute value of a number in is defined as and is denoted by . For each number in , . Then we say The non-Newtonian distance between two numbers and is defined by . Similarly, by taking into account the definition for -generator in (3), one can conclude that the equality holds for all .
Now, we give a new type calculus for non-Newtonian complex terms, denoted by -calculus, which is a branch of non-Newtonian calculus. From now on we will use -calculus type with respect to two arbitrarily selected generator functions.
Suppose that and are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics (-arithmetic, -arithmetic). The sets and are complete ordered fields and -generator generates -arithmetics, respectively. Definitions given for -arithmetic are also valid for -arithmetic.
The important point to note here is that -arithmetic is used for arguments and -arithmetic is used for values; in particular, changes in arguments and values are measured by -differences and -differences, respectively. The operators of this calculus type are applied only to functions with arguments in and values in . The -limit of a function at an element in is, if it exists, the unique number in such that for every sequence of arguments of distinct from , if is -convergent to , then -converges to and is denoted by . That is, A function is -continuous at a point in if and only if is an argument of and . When and are the identity function , the concepts of -limit and -continuity are identical with those of classical limit and classical continuity.
The isomorphism from -arithmetic to -arithmetic is the unique function (iota) that possesses the following three properties.(i) is one to one.(ii) is from onto .(iii)For any numbers and in , It turns out that for every in and that for every integer . Since, for example, , it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.
2.2. Non-Newtonian Complex Field and Some Inequalities
Let and be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair is called a -point. The set of all -points is called the set of -complex numbers and is denoted by ; that is, Define the binary operations addition and multiplication of -complex numbers and as follows: where and .
Theorem 4 (see ). is a field.
Following Grossman and Katz , we can give the definition of -distance and some applications with respect to the -calculus which is a kind of calculi of non-Newtonian calculus.
The -distance between two arbitrarily elements and of the set is defined by Up to now, we know that is a field and the distance between two points in is computed by the function , defined by (8).
Definition 5 (see ). Given a sequence of -complex numbers, the formal notation is called an infinite series with -complex terms, or simply complex -series. Also, for integers , the finite -sums are called the partial sums of complex -series. If the sequence -converges to a complex number then we say that the series -converges and write . The number is then called the -sum of this series. If -diverges, we say that the series -diverges, or that it is -divergent.
Remark 6. Given a sequence of -real numbers , the formal notation is called an infinite non-Newtonian series with real terms. Also, for integers , the finite sums are called the partial sums of the -series. If the sequence -converges to a real number then we say that the series -converges and write . The number is then called the sum of this series. If -diverges, we say that the -series is -divergent.
Proposition 7 (see ). For any . Then the following statements hold. (i). (-triangle inequality)(ii).(iii)Let and for . Then,
Folllowing Tekin and Başar , we can give the -norm and next derive some required inequalities in the sense of non-Newtonian complex calculus.
Let be an arbitrary element. The distance function is called -norm of and is denoted by . In other words, where and . Moreover, since for all we have which is the induced metric from norm.
Definition 8 (see , complex conjugate). Let . We define the -complex conjugate of by . Conjugation changes the sign of the imaginary part of but leaves the real part the same. Thus,
Theorem 9 (see ). is a complete metric space, where is defined by (8).
Corollary 10 (see ). is a Banach space with the -norm defined by ; .
3. Non-Newtonian Infinite Matrices
A non-Newtonian infinite matrix of non-Newtonian complex numbers is a double sequence of complex numbers defined by a function from the set into the complex field , where denotes the set of natural numbers, that is, . The complex number denotes the value of the function at and is called the entry of the matrix in the ith row and jth column.
The addition and scalar multiplication of the infinite matrices and are defined by where the elements , , , are in and is a non-Newtonian scalar in . The product of the infinite matrices and is defined by provided that the series on the right hand side of (15) -converge for all , where denotes the entry of the matrix in the ith row and jth column. For simplicity in notation, here and in what follows, the summation without limits runs from to . On the other hand, the series on the right hand side of (15) -converges if and only if are convergent classically for all . However the series (15) may -diverge for some, or all, values of ; the product of the infinite matrices may not exist.
Definition 11 (see ). Consider the following system of an infinite number of linear equations in infinitely many unknown elements by for all . If we construct a non-Newtonian infinite matrix with the coefficients of the unknowns and denote the -vectors of unknowns and constants by and , then the above sum can be expressed in matrix form as . Also , where is called -unit matrix and is defined by
A very important application of infinite matrices is used in the theory of summability of divergent sequences and series which is considered based on non-Newtonian mean in next chapter. A simple example of the non-Newtonian Cesaro mean, denoted by -Cesaro mean of order one, which is the analog of the well-known method of summability given below.
Example 12 (Cesàro mean). Define the matrix by If we choose the generator functions as and the calculus is bigeometric calculus [14, 15], then we obtain an infinite matrix with complex terms as follows:where and is a logarithmic number. The important point to note here is that the infinite matrix can be obtained in a similar way by using different generator functions above mentioned.
The -zero matrix is the matrix whose entries are all equal to . Thus, it is obvious that . But, as classical, does not imply or . Further, the conjugate of a complex matrix is the matrix where is the conjugate of the complex number in Definition 8.
3.1. Non-Newtonian Matrix Transformations
Let , and be an infinite matrix of non-Newtonian complex numbers for all and . Then, we say that defines a matrix mapping from into and denote it by writing , if for every sequence the sequence , the -transform of , exists and is in , where and, in this way, we transform the sequence , with and , into the sequence by for all and . Thus, if and only if the series on the right side of (21) -converges for each and every , and we have for all . On the other hand, we say if and only if the series and are convergent classically for all . A sequence is said to be -summable to if -converges to which is called as the - of . We denote the nth row of a matrix by for all ; that is, for all . Following Başar , we give some lines about ordinary and absolute summability of non-Newtonian complex numbers.
Let be an infinite matrix of non-Newtonian complex numbers throughout. We define two kinds of summability: ordinary and absolute summability, as shortly mentioned, below.
(a) Ordinary Summability. A sequence is said to be summable to a if the - of is for all ; that is, which implies that in classical mean for each . The matrix defines a summability method or a matrix transformation by (21).
(b) Absolute Summability. A sequence is said to be absolutely summable with index to a number if the series on the right hand side of (21) -converge for each and
The Cesàro transform of a sequence is given by , where the Cesàro method of one order is given by Example 12. Now, following Example 12, we may state the Cesàro summability with respect to the non-Newtonian calculus which is analogous to the classical Cesàro summable.
Example 13. Suppose that is an infinite sequence defined by where . One can easily conclude that . Then, since for all , . This means that the -divergent sequence is -summable to .
Tekin and Başar  have introduced the sets , , and of all bounded, convergent, null, and absolutely -summable sequences over the complex field which correspond to the sets and over the complex field , respectively. That is to say that It is not hard to show that the sets , , , and are the subspaces of the space . This means that , , , and are classical sequence spaces over the field and complete metric spaces with corresponding metrics.
Quite recently, Kadak  have introduced the sets , , and consisting of the sets of all bounded, convergent, and null series based on the non-Newtonian calculus, as follows:
Theorem 14 (see ). The following statements hold.(a)The sets , , , and ; are sequence spaces.(b)Let denote any of the spaces , , and and , . Define on the space by . Then, is a complete metric space.(c)The spaces , , and are Banach spaces with the norm defined by (d)The space is Banach spaces with the norm defined by
Theorem 15 (see ). Let denote any of the spaces , , and , and , . Define on the space by for arbitrarily chosen operators and corresponding function . Then, is a complete metric space.
Corollary 16 (see ). The spaces , , and are Banach spaces with the norm defined by
Theorem 17 (see ). Let be defined on the space by where , , and . Then, is a complete metric space.
Firstly, we give the alpha-, beta-, and gamma-duals of a set which are, respectively, denoted by , , and , as follows: where is the coordinatewise product of -complex numbers and for all . Then is called beta-dual of or the set of all convergence factor sequences of in . Firstly, we give a remark concerning with the -convergence factor sequences.
Theorem 18 (see ). The following statements hold. (a).(b).
Theorem 19 (see ). The following statements hold. (a).(b), , , .(c), , , .
Now, we give the characterizations of some matrix classes and state the necessary and sufficient condition on non-Newtonian matrix transformations by using the results given on Köthe-Toeplitz duals in .
Theorem 20. The following statements hold: (i) if and only if (ii) if and only if (33) holds.(iii) if and only if (33) holds.(iv) if and only if
Proof. Since the proof can also be obtained in the similar way for other cases, to avoid the repetition of the similar statements, we prove only case (i).
Suppose that condition (33) holds and . In this situation, since for every fixed , the -transform of exists. Taking into account the hypothesis, one can easily observe that which leads us to the fact that , as desired.
Conversely, suppose that . Put and observe that is a sequence of bounded linear operators on such that . Hence the results are obtained similarly from an application of Banach-Steinhaus theorem in classical mean.
Example 21. Let and define the matrix by for all . Then holds for otherwise . By taking into account , we obtain for all . This shows, by (i) of Theorem 20, that .
We state and prove the Kojima-Schur theorem which gives the necessary and sufficient conditions on an infinite matrix with respect to the non-Newtonian calculus, that maps the space into itself. A matrix satisfying the conditions of the Kojima-Schur theorem is called a conservative matrix or convergence preserving matrix.
Theorem 22 (Kojima-Schur). if and only if (33) holds, and there exist such that
Proof. Suppose that the conditions (33), (38), and (39) hold and with as . Then, since for each , the -transform of exists. In this situation, the equality
holds for each . In (40), since the terms on the right hand side tend to by (38) and the second term on the right hand side tends to by (39) as , in the sense of -limit, we have
Hence, ; that is the conditions are sufficient.
Conversely, suppose that . Then exists for every . By and , we denote the sequences such that for , and and . The necessity of the conditions (38) and (39) is immediate by taking and , respectively. Since , the necessity of the condition (33) is obtained from Theorem 20(i).
Theorem 23. if and only if (33) holds and there exists such that for each . If , then and .
Proof. Suppose that (33) and (42) hold. Then there exists an for and such that
for all . Since
for , by (42), one can see that and . Let . Then, one can choose a for such that for each fixed . Additionally, since , as by (42), we have , as for each fixed . That is to say that. Hence, there exists an such that for all . Thus, since
for all , the series are -convergent for each and , as . This means that .
Conversely, let and let . Then, since exists and the inclusion holds, the necessity of (33) is trivial by (iii) of Theorem 20. Now, if we take the sequence , then holds for each fixed ; that is, condition (42) is also necessary. Thus, the proof is completed.
As an easy consequence of Theorem 23, we have the following corollary.
Corollary 24. if and only if (33) holds and (42) also holds with for all .
Example 25. Let and . The Cesaro means of order r is defined by the matrix as Taking we obtain an infinite matrix as follows: One can easily conclude that for all and (33) holds. On the other hand, so (42) also holds with for all . Therefore .
A matrix satisfying the conditions of the Silverman-Toeplitz theorem is called a Toeplitz matrix or regular matrix. By , we denote the class of Toeplitz matrices. Now, we may give the corollaries characterizing the classes of .
Corollary 26 (Silverman-Toeplitz Theorem). if and only if (33) holds and (38) and (39) also hold with for all and , respectively.
Example 27. Example 25 can be given as an example of Silverman-Toeplitz theorem. Because the conditions (33) and (38) hold with . Furthermore we have and (39) also holds.
Theorem 28. if and only if
Proof. Let and . Then, the series -converges to for each fixed , since exists. Hence, for all . Define the sequence by for all . Then, which yields for all that
Furthermore we obtain
Conversely, suppose that (49) holds and . Then, since for each , exists. Therefore, one can observe, by using condition (49), that which means that , as desired.
Theorem 29. if and only if (38) holds, and
Proof. Let with and for all . Define the infinite matrix by for all . Suppose that . Then, exists for every and is in . Since this also holds for for each fixed , the necessity of (38) is clear. Consider the following relation obtained from mth-partial sums of the series by applying Abel's partial summation. In this situation, the equalities