## Tbilisi Mathematical Journal

### Double absolute indexed matrix summability with its applications

#### Abstract

The well established theory of summability of simple series has been brought to a high degree of development; however the extension of this theory to multiple series is still in its infancy. As regards to the double series, in the proposed paper a result on absolute indexed matrix summability with an additional parameter of doubly infinite lower triangular matrix has been established that generalizes a theorem of E. Savaş and B. E. Rhoades [10] (see E. Savaş and B. E. Rhoades, Double absolute summability factor theorems and applications, Nonlinear Anal. 69 (2008), 189-200). Furthermore, some concluding remarks and applications are presented in support of our result.

#### Article information

Source
Tbilisi Math. J., Volume 11, Issue 4 (2018), 1-18.

Dates
Accepted: 18 June 2018
First available in Project Euclid: 4 January 2019

https://projecteuclid.org/euclid.tbilisi/1546570881

#### Citation

Jena, B. B.; Paikray, S. K.; Misra, U. K. Double absolute indexed matrix summability with its applications. Tbilisi Math. J. 11 (2018), no. 4, 1--18. https://projecteuclid.org/euclid.tbilisi/1546570881

#### References

• A. A. Das, B. B. Jena, S. K. Paikray and R. K. Jati, Statistical Deferred Weighted Summability and Associated Korovokin-type Approximation Theorem, Nonlinear Sci. Lett. A, 9(3) (2018), 238–245.
• B. B. Jena, S. K. Paikray and U. K. Misra, A Tauberian theorem for double Cesàro summability method Int. J. Math. Math. Sci. 2016 (2016), Article ID, 1–4.
• B. B. Jena, S. K. Paikray and U. K. Misra, Inclusion theorems on general convergence and statistical convergence of - summability using generalized Tauberian conditions, Tamsui Oxf. J. Inf. Math. Sci. 31 (2017), 101–115.
• B. B. Jena, Vandana, S. K. Paikray and U. K. Misra, On Generalized Local Property of $| A;\delta| _ {k}$-Summability of Factored Fourier Series, Int. J. Anal. Appl. 16 (2018), 209–221.
• B. B. Jena, S. K. Paikray and U. K. Misra, Statistical deferred Cesaro summability and its applications to approximation theorems, Filomat 32 (2017), 1–13.
• B. B. Jena, L. N. Mishra, S. K. Paikray and U. K. Misra, Approximation of Signals by General Matrix Summability with Effects of Gibbs Phenomenon, Bol. Soc. Paran. Mat. (2018), doi:10.5269/bspm.v38i6.39280.
• S. K. Paikray, U. K. Misra and N. C. Sahoo, Trangular Matrix Summability of a series, African Jour. of Math. and Comput. Sci. Res. 4 (2011), 164–169.
• T. Pradhan, S. K. Paikray, B. B. Jena and H. Dutta, Statistical deferred weighted $\mathcal{B}$-summability and its applications to associated approximation theorems, J. Inequal. Appl. 2018 (2018), 1–21, Article Id: 65.
• E. Savaş, On generalized absolute summability factors, Nonlinear Anal. 68 (2008), 229–234.
• E. Savaş and B. E. Rhoades, Double absolute summability factor theorems and applications, Nonlinear Anal. 69 (2008), 189–200.
• H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, A certain class of weighted statistical convergence and associated Korovkin type approximation theorems for trigonometric functions, Math. Methods Appl. Sci. 41 (2018), 671–683.
• H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. (RACSAM) 112 (2018), 1487–1501.
• H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, Deferred weighted A-statistical convergence based upon the (p, q)-Lagrange polynomials and its applications to approximation theorems, J. Appl. Anal. 24 (2018), 1–16.
• H. M. Srivastava, M. Mursaleen and A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Model. 55 (2012), 2040–2051.