Double Sequence Spaces by Means of Orlicz Functions

Abstract

We define some classes of double entire and analytic sequences by means of Orlicz functions. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.

1. Introduction and Preliminaries

Of the definitions of convergence commonly employed for double series, only that due to Pringsheim permits a series to converge conditionally. Therefore, in spite of any disadvantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series. Chief among the reasons why the theory of double sequences, under the Pringsheim definition of convergence, present difficulties not encountered in the theory of simple sequences, is the fact that a double sequence may converge without being a bounded function of and . Thus it is not surprising that many authors in dealing with the convergence of double sequences should have restricted themselves to the class of bounded sequences or in dealing with the summability of double series to the class of series for which the function whose limit is the sum of the series is a bounded function of and . Without such a restriction, peculiar things may sometimes happen; for example, a double power series may converge with partial sum unbounded at a place exterior to its associated circles of convergence. Nevertheless there are problems in the theory of double sequences and series where this restriction of boundedness as it has been applied is considerably more stringent than need be. The initial works on double sequences are found in Bromwich [1]. Later on, it was studied by Hardy [2], Móricz [3], Móricz and Rhoades [4], Basarir and Sonalcan [5], and many others. Hardy [2] introduced the notion of regular convergence for double sequences. Mursaleen and Mohiuddine [6, 7] have characterized four dimensional matrix transformations between double sequences . A good account of the study of double sequences can be found in most recent monograph by Mursaleen and Mohiuddine [8]. More recently, Altay and Başar [9] have defined the spaces , , , , , and of double sequences consisting of all double series whose sequence of partial sums is in the spaces , , , , , and , respectively, and also examined some properties of these sequence spaces and determined the -duals of the spaces , , and and the -duals of the spaces and of double series. Now, recently, Başar and Sever [10] have introduced the Banach space of double sequences corresponding to the well-known space of single sequences and examined some properties of the space . By the convergence of a double sequence we mean the convergence in the Pringsheim sense; that is, a double sequence has Pringsheim limit (denoted by ) provided that given there exists such that whenever ; see [11]. We will write more briefly as -convergent. The double sequence is bounded if there exists a positive number such that for all and .

An Orlicz function is a continuous, nondecreasing, and convex function such that , for and as . Lindenstrauss and Tzafriri [12] used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences ; then which is called as an Orlicz sequence space. Also is a Banach space with the norm Also, it was shown in [13] that every Orlicz sequence space contains a subspace isomorphic to . The - condition is equivalent to , for all with . An Orlicz function can always be represented in the following integral form: where is known as the kernel of and is right differentiable for , , and ; is nondecreasing and as .

Let be a linear metric space. A function is called paranorm, if(1), for all ,(2), for all ,(3), for all ,(4)if is a sequence of scalars with and is a sequence of vectors with , then .

A paranorm for which implies is called total paranorm and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [14], Theorem , p.183). For more details about sequence spaces see [13, 15–21].

A complex sequence, whose th term is , is denoted by . Let be the set of all finite sequences. A sequence is said to be analytic if . The vector space of all analytic sequences will be denoted by . A sequence is called entire sequence if  . The vector space of all entire sequences will be denoted by .

The notion of difference sequence spaces was introduced by Kızmaz [22], who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak [23] by introducing the spaces , , and .

Let , be nonnegative integers; then for we have sequence spaces where and for all , which is equivalent to the following binomial representation: Taking , we get the spaces which were studied by Et and Çolak [23]. Taking , we get the spaces which were introduced and studied by Kızmaz [22].

Let denote the space of all complex double sequences . The space consisting of all those sequences in such that for some arbitrary fixed is denoted by and is known as double Orlicz space of entire sequences. The space is a metric space with the metric for all and in .

The space consisting of all those sequences in such that for some arbitrarily fixed is denoted by and is known as double Orlicz space of analytic sequences.

A double sequence space is said to be solid or normal if whenever and for all sequences of scalars with (see [18]).

The following inequality will be used throughout the paper. Let be a double sequence of positive real numbers with and let . Then for the factorable sequences and in the complex plane, we have Let be a sequence of Orlicz functions, let be a bounded sequence of positive real numbers, let be a sequence of strictly positive real numbers, and let be locally convex Hausdorff topological linear space whose topology is determined by a set of continuous seminorms . The symbols   , denote the space of all double analytic and double entire sequences, respectively, defined over . In this paper we define the following sequence spaces: The main aim of this paper is to introduce some double entire sequence spaces and defined by a sequence of Orlicz functions and study some topological properties and inclusion relation between these spaces.

2. Main Results

Theorem 1. Let be a sequence of Orlicz functions, let be a bounded sequence of positive real numbers, and let be a sequence of strictly positive real numbers; then the spaces and are linear spaces over the field of complex numbers .

Proof. Let and . In order to prove the result, we need to find some such that Since , there exist some positive and such that Since is a nondecreasing convex function, is a seminorm and is linear and so, by using inequality (6), we have Take such that : Hence This proves that is a linear space. Similarly, we can prove is a linear space.

Theorem 2. Let be a sequence of Orlicz functions, let be a bounded sequence of positive real numbers, and let be a sequence of strictly positive real numbers; then the space is a paranormed space with paranorm defined by where .

Proof. Clearly and , where is the zero sequence of . For , there exist such that Suppose that ; then Hence Thus we have . Hence satisfies the triangle inequality. Now, where . Hence is a paranormed space.

Theorem 3. If and are two sequences of Orlicz functions, then

Proof. Let . Then there exist and such that Let . Then we have by (20). Therefore, .

Theorem 4. Let . Then we have the following inclusions:(i),(ii).

Proof. Let . Then we have Since is nondecreasing convex function and is a seminorm, we have Therefore, as . Hence, . This completes the proof of (i). Similarly, we can prove (ii).

Theorem 5. Let and let be bounded. Then .

Proof. Let . Then Let and . Since , we have . Take . Define where and . It follows that and . Since , then . Thus, Therefore, Hence . From (24), we get .

Theorem 6. (i) Let . Then . (ii) Let . Then .

Proof. (i) Let . Then Since , From (28) and (29) it follows that . Thus .
(ii) Let for each and and let . Then Since , we have This implies that . Therefore, .

Theorem 7. Suppose ; then .

Proof. Let . Then we have But , by our assumption, implies that Then and .

Theorem 8. is solid.

Proof. Let ; then Let be a double sequence of scalars such that for all . Then we have and this completes the proof.

Corollary 9. is monotone.

Proof. It is obvious.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.