Deferred Cesàro and deferred Euler equi-statistical convergence and its applications to Korovkin-type approximation theorem

Abstract

The present paper emphasises on equi-statistical convergence, pointwise statistical convergence and uniform statistical convergence for a sequence of real-valued functions by using deferred Cesàro and deferred Euler statistical convergence and obtain various implicative results with supporting examples. We make an effort to demonstrate Korovkin-type approximation theorem via deferred Cesàro and deferred Euler equi-statistical convergence. We also present an example which shows that our Korovkin-type theorem is powerful than its classical version. Further, we study rates of deferred Cesàro and deferred Euler equi-statistical convergence via modulus of continuity.

Keywords:

• Deferred Cesaro mean
• deferred Euler mean
• statistical convergence
• rate of convergence
• approximation theorem

Acknowledgments

The authors are grateful to the anonymous referees for their valuable suggestions which improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Kavita Saini

Kavita Saini has passed B.Sc. from University of Jammu (J&K) in 2015. She has completed M.Sc. (Mathematics) from Shri Mata Vaishno Devi University, Katra (J&K), India, in 2017. Currently pursuing Ph.D. under supervision of Dr Kuldip Raj at School of Mathematics, Shri Mata Vaishno Devi University, Katra (J&K), India. Her research interests include Functional Analysis, Sequence Spaces and Summability Theory.

Kuldip Raj

Kuldip Raj has received his post graduation and Ph.D. degree in Mathematics from University of Jammu in 1992 and 1999, respectively. Currently, he is working as an Assistant Professor at School of Mathematics, Shri Mata Vaishno Devi University, Katra (J&K), India. He is teaching undergraduate and post graduate students for over 19 years. He has published a number of research papers in journals of national and international repute. His research area is Functional Analysis, Operator Theory, Sequence Spaces and Summability Theory.

M. Mursaleen

Mohammad Mursaleen is currently a Principal Investigator for an SERB Core Research Grant at the Department of Mathematics, Aligarh Muslim University. He is also Visiting Professor at China Medical University, Taiwan, since January 2019. He has served as Lecturer to Full Professor at Aligarh Muslim University since 1982; and as Chair of the Department of Mathematics from 2015 to 2018. He has published more than 350 research papers in the field of Summability, Sequence Spaces, Approximation Theory, Fixed Point Theory, Measures of noncompactness. He has also published nine books and completed several national and international projects, in several countries. Besides several master's students, he has guided 21 Ph.D. students. He has served as a reviewer for various international scientific journals and is a member of editorial boards for many international scientific journals. He has many academic visits of a number of foreign universities/ institutions in several countries. He is in the list of Highly Cited Researchers for the year 2019 of Thomson Reuters (Web of Science).