Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems
The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. <b>41</b> (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space a...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-03-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/11/4/448 |
| _version_ | 1798043519839895552 |
|---|---|
| author | Hari Mohan Srivastava Bidu Bhusan Jena Susanta Kumar Paikray Umakanta Misra |
| author_facet | Hari Mohan Srivastava Bidu Bhusan Jena Susanta Kumar Paikray Umakanta Misra |
| author_sort | Hari Mohan Srivastava |
| collection | DOAJ |
| description | The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. <b>41</b> (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions. |
| first_indexed | 2024-04-11T22:50:15Z |
| format | Article |
| id | doaj.art-b1b2c607a1eb4c3ba3785fd0d25bf72b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T22:50:15Z |
| publishDate | 2019-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-b1b2c607a1eb4c3ba3785fd0d25bf72b2022-12-22T03:58:36ZengMDPI AGSymmetry2073-89942019-03-0111444810.3390/sym11040448sym11040448Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation TheoremsHari Mohan Srivastava0Bidu Bhusan Jena1Susanta Kumar Paikray2Umakanta Misra3Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, IndiaDepartment of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, IndiaDepartment of Mathematics, National Institute of Science and Technology, Palur Hills, Golanthara, Odisha 761008, IndiaThe concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. <b>41</b> (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.https://www.mdpi.com/2073-8994/11/4/448statistical convergence<i>P</i>-convergentstatistically and relatively modular deferred-weighted summabilityrelatively modular deferred-weighted statistical convergenceKorovkin-type approximation theoremmodular spaceconvex space<i>N</i>-quasi convex modular<i>N</i>-quasi semi-convex modular |
| spellingShingle | Hari Mohan Srivastava Bidu Bhusan Jena Susanta Kumar Paikray Umakanta Misra Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems Symmetry statistical convergence <i>P</i>-convergent statistically and relatively modular deferred-weighted summability relatively modular deferred-weighted statistical convergence Korovkin-type approximation theorem modular space convex space <i>N</i>-quasi convex modular <i>N</i>-quasi semi-convex modular |
| title | Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems |
| title_full | Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems |
| title_fullStr | Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems |
| title_full_unstemmed | Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems |
| title_short | Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems |
| title_sort | statistically and relatively modular deferred weighted summability and korovkin type approximation theorems |
| topic | statistical convergence <i>P</i>-convergent statistically and relatively modular deferred-weighted summability relatively modular deferred-weighted statistical convergence Korovkin-type approximation theorem modular space convex space <i>N</i>-quasi convex modular <i>N</i>-quasi semi-convex modular |
| url | https://www.mdpi.com/2073-8994/11/4/448 |
| work_keys_str_mv | AT harimohansrivastava statisticallyandrelativelymodulardeferredweightedsummabilityandkorovkintypeapproximationtheorems AT bidubhusanjena statisticallyandrelativelymodulardeferredweightedsummabilityandkorovkintypeapproximationtheorems AT susantakumarpaikray statisticallyandrelativelymodulardeferredweightedsummabilityandkorovkintypeapproximationtheorems AT umakantamisra statisticallyandrelativelymodulardeferredweightedsummabilityandkorovkintypeapproximationtheorems |