On the last five decades the interest of the study of positive approximation processes have emerged with growing evidence. A special place is occupied by the in-depth study of classical operators. The most eloquent example is Bernstein operator which represents a permanent challenge for the researches in the mentioned field. However, in this synthesis we focused on presenting a class of operators introduced by G.C. Jain in the 1970s that have long been in a shadowy cone. In recent years many papers have appeared about their properties and many generalizations have been analyzed. In our approach, there is no question of an exhaustive treatment, but only of collecting some published results that prove the importance of this class through the generous possibilities offered by the approximation of signals from different function spaces.

## Abstract

## Authors

## Keywords

### References

See the expanding block below.

## Paper coordinates

O. Agratini, *A stop over Jain operators and their generalizations, *Annals of West University of Timisoara – Mathematics and Computer Science, 56 (2018) no. 2, pp. 28-42, doi.org/10.2478/awutm-2018-0014

https://content.sciendo.com/view/journals/awutm/56/2/article-p28.xml

## About this paper

##### Publisher Name

Sciendo

##### Print ISSN

##### Online ISSN

1841-3307

##### Google Scholar Profile

soon

[1] U. Abel, O. Agratini, *Asymptotic behaviour of Jain operators*, Numer. Algor. 71 (2016), 553-565.

[2] O. Agratini, *Approximation properties of a class of linear operators,* Math. Meth. Appl. Sci. 36 (2013), 2353-2358.

[3] O. Agratini, *On an approximation process of integral type*, Appl. Math. Comp. 236 (2014), 195-201.

[4] O. Agratini, *Uniform approximation of some classes of linear positive operators expressed by series*, Applicable Analysis 94 (2015), No. 8, 1662-1669.

[5] G. Bascanbaz-Tunca, M. Bodur, D. Soylemez, *On Lupas-Jain operators,* Stud. Univ. Babes-Bolyai Math. 63 (2018), No. 4, 525-537.

[6] J. Bustamante, L. Morales de la Cruz, *Korovkin type theorems for weighted approximation*, Int. Journal of Math. Analysis 1 (2007), No. 26, 1273-1283.

[7] J.S. Connor, *The statistical and strong p-Cesaro convergence of sequences*, Analysis 8 (1988), 47-63.

[8] E. Deniz, *Quantitative estimates for Jain-Kantorovich operators*, Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat. 65 (2016), No. 2, 121-132.

[9] A. Farcas, *An aymptotic formula for Jain’s operators,* Stud. Univ. Babes-Bolyai Math. 57 (2012), No. 4, 511-517.

[10] H. Fast, *Sur la convergence statistique*, Colloq. Math. 2 (1951), 241-244.

[11] J.A. Fridy, M.K. Khan, *Statistical extensions of some classical Tauberian theorems*, Proceedings of American Math. Soc. 128 (2000), No. 8, 2347-2355.

[12] J.A. Fridy, H.I. Miller, *A matrix characterization of statistical convergence,* Analysis 11 (1991), 59-66.

[13] A.D. Gadzhiev, *Theorems of Korovkin type*, Math. Notes 20 (1976), No. 5, 995-998.

[14] A.D. Gadjiev, C. Orhan, *Some approximation theorems via statistical convergence,* Rocky Mountain J. Math. 32 (2002), No. 1, 129-138.

[15] V. Gupta, R.P. Agarwal, *Convergence Estimates in Approximation Theory,* Springer, 2014.

[16] V. Gupta, M.A. Noor, *Convergence of derivatives for certain mixed Szasz-Beta operators,* J. Math. Anal. Appl. 321 (2006), 1-9.

[17] G.C. Jain, *Approximation of functions by a new class of linear operators*, J. Australian Math. Soc. 13 (1972), No. 3, 271-276.

[18] H. Johnen, *Inequalities connected with moduli of smoothness*, Matematicki Vesnik 9 (24) (1972), 289-303.

[19] A. Lupas, *The approximation by some positive linear operators, In: Proceedings of the International Dortmund Meeting on Approximation*

*Theory*, (M.W. Muller et al., eds.), Mathematical Research, Akademie Verlag, Berlin, 86 (1995), 201-229.

[20] S.M. Mazhar, V. Totik, *Approximation by modified Szasz operators*, Acta Sci. Math. 49 (1985), 257-269.

[21] G.M. Mirakjan, *Approximation of functions with the aid of polynomials*, Dokl. Akad. Nauk SSSR 31 (1941), 201-205 (in Russian).

[22] A. Olgun, F. Tasdelen, A. Erencin, *A generalization of Jain’s operators*, Appl. Math. Comp. 266 (2015), 6-11.

[23] J. Peetre, *A Theory of Interpolation of Normed Spaces,* Notas de Matematica Instituto de Matematica Pura e Aplicada, Rio de Janeiro, 39 (1968), 1-86.

[24] R.S. Phillips, *An inversion formula for Laplace transforms and semigroups of linear operators*, Ann. Math. Second Ser. 59 (1954), 325-356.

[25] H. Steinhaus, *Sur la convergence ordinaire et la convergence asymptotique,* Colloq. Math. 2 (1951), 73-74.

[26] O. Szasz, *Generalization of S. Bernstein’s polynomials to the infinite interval*, J. Res. Nat. Bur. Standards 45 (1950), 239-245.

[27] S. Tarabie, *On Jain-Beta linear operators*, Appl. Math. Inf. Sci. 6 (2012), No. 2, 213-216.

[28] S. Umar, Q. Razi, *Approximation of function by generalized Szasz operators*, Commun. Fac. Sci. de l’Universite d’Ankara, Serie A1: Mathematique 34 (1985), 45-52.

### Publications

## Methods of Newton and Newton-Krylov type

## Solutions with a prescribed interval of positivity for differential systems with nonlocal conditions

### Publication Fields

- (original) (295)
- (preprint) (16)
- (proceedings) (24)
- (survey) (2)
- Announcements (27)
- Approximation Theory (11)
- book chapter (3)
- Books (4)
- Chebyshev method (4)
- Convergence orders (8)
- divided differences (10)
- eigenvalue/eigenvector problems (8)
- finite differences (FD) (2)
- Finite element (FEM) (9)
- Fixed point theory (13)
- global random walk (14)
- history (3)
- inexact/perturbed iterations (11)
- inverse interpolation (9)
- ISI/JCR (96)
- iterative methods (33)
- Krylov methods (4)
- linear systems in Rn (5)
- local convergence (19)
- metals and alloys casting (12)
- Meteorology (5)
- Monte Carlo simulations (3)
- Newton method (20)
- nonlinear equations in Banach spaces (6)
- nonlinear equations in R (10)
- nonlinear systems in Rn (11)
- Numerical Analysis (89)
- Numerical Modeling (94)
- ODEs (2)
- Optimization (4)
- paper (340)
- PDEs (7)
- Schulz type iterations (2)
- secant/chord method (5)
- semilocal convergence (8)
- spectral methods (2)
- Steffenssen methods (7)
- successive approximations (5)
- talk (4)
- Time Series (13)
- Uncategorized (3)