Abstract
In this article, we introduce the class of sequence of functions of Cesaŕo summable relative uniform difference sequence of functions. We have studied the topological properties of . We also obtain the necessary and sufficient condition to characterize the matrix classes .
1. Introduction
Throughout the study, denote the classes of all relative uniform sequence space, Cesaŕo summable relative uniform difference sequence space, and bounded relative uniform sequence space, respectively.
Moore in 1910 introduced the notion of uniform convergence of a sequence of functions relative to a scale function. Chittenden [1–3] gave the detailed definition of the notion as follows.
Definition 1 (see [1]). A sequence of single-valued, real-valued functions of a variable ranging over a compact subset of real numbers is said to be relatively uniformly convergent on w. r. t. a scale function in case there exist a limiting function and scale function defined on and for every , an integer such that for every and for all ,
The notion was further discussed from various aspects by Demirci et al. [4], Demirci and Orhan [5], Devi and Tripathy [6], and many others.
Example 1. Let be a real number. Consider the sequence of functions , for all defined by
This sequence of functions does not converge uniformly to 0 on . However, converges to 0 uniformly with respect to the scale function defined by
Kizmaz [7] defined the difference sequence spaces as follows: for where .
These sequence spaces are Banach space under the norm
The notion was further studied from different aspects by many others [8–13].
The Cesaŕo sequence space were introduced by Shiue [14], and it has been shown that ; the inclusion is strict for . Further, the Cesaŕo sequence spaces and of nonabsolute type were defined by Ng and Lee [15, 16]. For a detail account of Cesaŕo difference sequence space, one may refer to [17–20].
Let be an infinite matrix of real or complex numbers. Then, transforms from the sequence space into the sequence space , if for each sequence that is , where provided that the infinite series converges for each .
Matrix transformation between sequence space was studied from different aspects by many others [21–25].
2. Definitions and Preliminaries
Definition 2. A sequence space is said to be solid or normal if implies , for all with , for all .
Definition 3. A sequence space is said to be monotone if it contains the canonical preimages of all its step spaces.
Remark 4. A sequence space is solid then, is monotone.
Definition 5. A sequence space is said to be symmetric if , for all , where is a permutation of , the set of natural numbers.
Definition 6. A sequence space is said to be convergence free if and together with , for all .
Definition 7. A sequence space is said to be a sequence algebra if whenever and belongs to , for all .
In this article we introduce the sequence space of Cesaŕo summable relative uniform difference sequence of functions and it is defined as follows:
where (ru) and
3. Main Results
We state the following result without proof.
Theorem 8. The sequence space is a normed linear space.
Theorem 9. The sequence space is a Banach space normed by
Proof. Let be a Cauchy sequence in where
Then,
For all ,
is a Cauchy sequence in w.r.t. for all .
is convergent in w.r.t. for all .
Let .
Similarly, .
From the above equations we get, for all , for all .
From (10) we have for all .
Similarly, for all and .
Since is not dependent on , we have
Evidently,
for all for all .
Therefore, for all for all .
Then, since is a linear space.
Theorem 10. The inclusion strictly holds.
Proof. The proof of the theorem is obvious and the strictness of the inclusion is shown in the following example.
Example 1. Let be a real number and Consider the sequence of real valued functions , for all , defined by
w.r.t. the scale function , for all xin D, but .
Hence, the inclusion is strict.
Theorem 11. The inclusion strictly holds.
Proof. The proof is obvious and the strictness of the inclusion is shown in the following example.
Example 2. Let be a real number and Consider the sequence of real valued functions , for all , defined by We have w.r.t. the scale function defined by but Hence, the inclusion is strict.
Theorem 12. The sequence space is not monotone.
Proof. The proof is shown in the following example.
Example 3. Let be a real number and Consider the sequence of real valued functions , for all , defined by
w.r.t. the scale function defined on D by .
Let be the preimage of defined by
One cannot get a scale function that makes .
Hence is not monotone.
Remark 13. The sequence space is not solid since is not monotone.
Theorem 14. The sequence space is not symmetric.
Proof. The proof of the theorem is shown with the help of the following example.
Example 4. Let us consider the sequence of functions considered in Example 3. Let be the rearrangement sequence of functions of defined by One cannot get a scale function that makes .
Hence, is not symmetric.
Theorem 15. The sequence space is not sequence algebra.
Proof. The proof of the theorem is shown in the following example.
Example 5. Let be a real number and Consider the sequences of real valued functions , and , for all defined by
We get that but one cannot get a scale function that makes .
Hence, is not sequence algebra.
Theorem 16. The sequence space is not convergence free.
Proof. The proof of the theorem follows from example below.
Example 6. Let be a real number and Consider the sequence of real valued functions , for all , defined by
Therefore, w.r.t. the constant scale function defined on by .
Let us consider another sequence of functions defined by
One cannot find a scale function that makes .
Hence, the sequence space is not convergence free.
3.1. Matrix Transformation between Sequence of Functions
In this section, we give certain matrix classes between the sequence of functions.
Theorem 17. if and only if .
Proof. Let and .
From the relation between dual and matrix map, we know that . Hence,
For , we have,
By the argument (10), we know that is absolutely convergent w.r.t. scale function .
Then,
We have, since
Conversely, we know that is a bounded linear function from to , so we can write,
We choose a sequence of functions defined by
We get w.r.t. scale function with the norm .
Putting the value of in equation (26) and letting the limit , we get
Theorem 18. if and only if (i)(ii)(iii) for each (iv)
Proof. Let us assume that the conditions (i)-(iv) hold true and let , i.e., (say).
We know from (i) that for each and , converges. It follows that converges.
For any , we have
Let , we get
Substituting this value in equation (31) and using conditions (ii) and (iv), we get
converges to w.r.t. the scale function Conversely, let . Then, , for all
Condition (i) can be proceed as same as shown in Theorem 17.
(ii) Let defined by
Then,
Since , we have
(iii) Let defined by
Then,
Since , we have
(iv) Let defined by
Then,
Since , we have
The following theorem is stated without proof and can be proceeded the same as in Theorem 18.
Theorem 19. if and only if (i)(ii)(iii) for each (iv)
4. Conclusions
In this article, we studied the concept of Cesaŕo summability from the aspects of relative uniform convergence of difference sequence of positive linear functions w.r.t. a scale function on a compact domain . The class of difference sequence of functions is introduced, and its properties like solid, monotone, symmetric, sequence algebra, and convergence free are discussed. We have also further introduced characterization of matrix classes of and .
Data Availability
No data were used to support to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was funded by the University of Jeddah, Saudi Arabia, under grant No. UJ-21-DR-93. The authors, therefore, acknowledge with thanks the university technical and financial support.