doi: 10.3934/mfc.2021044
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On a special class of modified integral operators preserving some exponential functions

Department of Computer Technologies, Division of Technology of Security of Informatics, Karabuk University, Karabuk, Turkey

Received  August 2021 Revised  December 2021 Early access January 2022

In the present paper, we consider a general class of operators enriched with some properties in order to act on $ C^{\ast }( \mathbb{R} _{0}^{+}) $. We establish uniform convergence of the operators for every function in $ C^{\ast }( \mathbb{R} _{0}^{+}) $ on $ \mathbb{R} _{0}^{+} $. Then, a quantitative result is proved. A quantitative Voronovskaya-type estimate is obtained. Finally, some applications are provided concerning particular kernel functions.

Citation: Gümrah Uysal. On a special class of modified integral operators preserving some exponential functions. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021044
References:
[1]

T. Acar, A. Aral and H. Gonska, On Szász-Mirakyan operators preserving $e^2ax$, $a>0$, Mediterr. J. Math., 14 (2017), Paper No. 6, 14 pp. doi: 10.1007/s00009-016-0804-7.  Google Scholar

[2]

T. Acar, M. Mursaleen and S. N. Deveci, Gamma operators reproducing exponential functions, Adv. Difference Equ., (2020), Paper No. 423, 13 pp. doi: 10.1186/s13662-020-02880-x.  Google Scholar

[3]

O. AgratiniA. Aral and E. Deniz, On two classes of approximation processes of integral type, Positivity, 21 (2017), 1189-1199.  doi: 10.1007/s11117-016-0460-y.  Google Scholar

[4]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics 17., Walter De Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.  Google Scholar

[5]

G. A. Anastassiou and S. G. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Springer, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1360-4.  Google Scholar

[6]

L. Angeloni and G. Vinti, A review on approximation results for integral operators in the space of functions of bounded variation, J. Funct. Spaces, 2016 (2016), Art. ID 3843921, 11 pp. doi: 10.1155/2016/3843921.  Google Scholar

[7]

P. M. Anselone and I. H. Sloan, Integral equations on the half line, J. of Integral Equations, 9 (1985), 3-23.   Google Scholar

[8]

A. Aral, On generalized Picard integral operators, Advances in Summability and Approximation Theory, (2018), 157–168. doi: 10.1007/978-981-13-3077-3_9.  Google Scholar

[9]

A. AralD. Cárdenas-Morales and P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 12 (2018), 861-872.  doi: 10.7153/jmi-2018-12-64.  Google Scholar

[10]

A. AralD. Inoan and I. Raşa, Approximation properties of Szász–Mirakyan operators preserving exponential functions, Positivity, 23 (2019), 233-246.  doi: 10.1007/s11117-018-0604-3.  Google Scholar

[11]

A. Aral, B. Yılmaz and E. Deniz, A new construction of Picard operators on the semi-real axis, (2018), to appear. Google Scholar

[12]

F. Barbieri, Approximation by moment kernels, (Italian), Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 308-328.   Google Scholar

[13]

C. Bardaro and I. Mantellini, Voronovskaja-type estimates for Mellin convolution operators, Results Math., 50 (2007), 1-16.  doi: 10.1007/s00025-006-0231-3.  Google Scholar

[14]

C. Bardaro and I. Mantellini, A quantitative Voronovskaya formula for Mellin convolution operators, Mediterr. J. Math., 7 (2010), 483-501.  doi: 10.1007/s00009-010-0062-z.  Google Scholar

[15]

C. Bardaro and I. Mantellini, Multivariate moment type operators: Approximation properties in Orlicz spaces, J. Math. Inequal., 2 (2008), 247-259.  doi: 10.7153/jmi-02-22.  Google Scholar

[16]

C. Bardaro, I. Mantellini, G. Uysal and B. Yılmaz, A class of integral operators that fix exponential functions, Mediterr. J. Math., 18 (2021), Paper No. 179, 21 pp. doi: 10.1007/s00009-021-01819-0.  Google Scholar

[17]

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications 9., Walter De Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110199277.  Google Scholar

[18]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2 (1952), 43-56.  doi: 10.1007/BF02591381.  Google Scholar

[19]

B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 14 (1970), 9-13.   Google Scholar

[20] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation Vol. 1: One-Dimensional Theory, Pure and Applied Mathematics, Vol. 40. Academic Press, New York-London, 1971.  doi: 10.1007/978-3-0348-7448-9.  Google Scholar
[21]

P. L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), 325-376.  doi: 10.1007/BF02649101.  Google Scholar

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183. doi: 10.1007/978-1-4613-9757-1_5.  Google Scholar

[23]

D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34 (2013), 819-844.  doi: 10.1080/01630563.2013.767833.  Google Scholar

[24]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Mathematical Foundations of Computing, 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.  Google Scholar

[25]

A. D. Gadžiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, (Russian), Dokl. Akad. Nauk SSSR, 218 (1974), 1001-1004.   Google Scholar

[26]

V. Gupta and V. K. Singh, Modified Post-Widder operators preserving exponential functions, Advances in Mathematical Methods and High Performance Computing, 41 (2019), 181-192.  doi: 10.1007/978-3-030-02487-1_10.  Google Scholar

[27]

V. Gupta and G. Tachev, On approximation properties of Phillips operators preserving exponential functions, Mediterr. J. Math., 14 (2017), Paper No. 177, 12 pp. doi: 10.1007/s00009-017-0981-z.  Google Scholar

[28]

A. Holhoş, The rate of approximation of functions in an infinite interval by positive linear operators, Stud. Univ. Babeş–Bolyai Math., 55 (2010), 133–142.  Google Scholar

[29]

A. Holhoş, Quantitative estimates of Voronovskaya type in weighted spaces, Results Math., 73 (2018), Paper No. 53, 11 pp. doi: 10.1007/s00025-018-0814-9.  Google Scholar

[30]

H. Karslı, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal., 85 (2006), 781-791.  doi: 10.1080/00036810600712665.  Google Scholar

[31]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.  Google Scholar

[32]

P. P. Korovkin, On convergence of linear positive operators in the spaces of continuous functions, (Russian), Doklady Akad. Nauk. SSSR (N.S.), 90 (1953), 961-964.   Google Scholar

[33]

P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corp., Delhi, 1960.  Google Scholar

[34]

A. Lupaş and M. Müller, Approximationseigenschaften der Gammaoperatoren, (German), Math. Z., 98 (1967), 208-226.  doi: 10.1007/BF01112415.  Google Scholar

[35]

R. G. Mamedov, The Mellin Transform and Approximation Theory, (Russian) "Elm", Baku, 1991.  Google Scholar

[36]

C. P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canadian J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8.  Google Scholar

[37]

I. P. Natanson, Theory of Functions of a Real Variable Vol. Ⅱ., Frederick Ungar Pub. Co., New York, 1961.  Google Scholar

[38]

R. S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. of Math., 59 (1954), 325-356.  doi: 10.2307/1969697.  Google Scholar

[39]

L. Rempulska and K. Tomczak, On some properties of the Picard operators, Arch. Math. (Brno), 45 (2009), 125-135.   Google Scholar

[40] L. L. Schumaker, Spline Functions: Basic Theory, 3$^rd$ edition, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618994.  Google Scholar
[41]

T. Świderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. (Prace Mat.), 40 (2000), 181-189.   Google Scholar

[42]

E. V. Voronovskaya, Determination of the asymptotic form of approximation of functions by the polynomials of S. N. Bernstein, Dokl. Akad. Nauk SSSR, A, (1932), 79–85. Google Scholar

[43]

E. Wachnicki and G. Krech, Approximation of functions by nonlinear singular integral operators depending on two parameters, Publ. Math. Debrecen, 92 (2018), 481-494.  doi: 10.5486/PMD.2018.8080.  Google Scholar

[44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series, Vol. 6. Princeton Univ. Press, Princeton, 1941.   Google Scholar
[45]

B. YılmazG. Uysal and A. Aral, Reconstruction of two approximation processes in order to reproduce $e^ax$ and $e^2ax$, $a>0$, J. Math. Inequal., 15 (2021), 1101-1118.  doi: 10.7153/jmi-2021-15-75.  Google Scholar

show all references

References:
[1]

T. Acar, A. Aral and H. Gonska, On Szász-Mirakyan operators preserving $e^2ax$, $a>0$, Mediterr. J. Math., 14 (2017), Paper No. 6, 14 pp. doi: 10.1007/s00009-016-0804-7.  Google Scholar

[2]

T. Acar, M. Mursaleen and S. N. Deveci, Gamma operators reproducing exponential functions, Adv. Difference Equ., (2020), Paper No. 423, 13 pp. doi: 10.1186/s13662-020-02880-x.  Google Scholar

[3]

O. AgratiniA. Aral and E. Deniz, On two classes of approximation processes of integral type, Positivity, 21 (2017), 1189-1199.  doi: 10.1007/s11117-016-0460-y.  Google Scholar

[4]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics 17., Walter De Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.  Google Scholar

[5]

G. A. Anastassiou and S. G. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Springer, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1360-4.  Google Scholar

[6]

L. Angeloni and G. Vinti, A review on approximation results for integral operators in the space of functions of bounded variation, J. Funct. Spaces, 2016 (2016), Art. ID 3843921, 11 pp. doi: 10.1155/2016/3843921.  Google Scholar

[7]

P. M. Anselone and I. H. Sloan, Integral equations on the half line, J. of Integral Equations, 9 (1985), 3-23.   Google Scholar

[8]

A. Aral, On generalized Picard integral operators, Advances in Summability and Approximation Theory, (2018), 157–168. doi: 10.1007/978-981-13-3077-3_9.  Google Scholar

[9]

A. AralD. Cárdenas-Morales and P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 12 (2018), 861-872.  doi: 10.7153/jmi-2018-12-64.  Google Scholar

[10]

A. AralD. Inoan and I. Raşa, Approximation properties of Szász–Mirakyan operators preserving exponential functions, Positivity, 23 (2019), 233-246.  doi: 10.1007/s11117-018-0604-3.  Google Scholar

[11]

A. Aral, B. Yılmaz and E. Deniz, A new construction of Picard operators on the semi-real axis, (2018), to appear. Google Scholar

[12]

F. Barbieri, Approximation by moment kernels, (Italian), Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 308-328.   Google Scholar

[13]

C. Bardaro and I. Mantellini, Voronovskaja-type estimates for Mellin convolution operators, Results Math., 50 (2007), 1-16.  doi: 10.1007/s00025-006-0231-3.  Google Scholar

[14]

C. Bardaro and I. Mantellini, A quantitative Voronovskaya formula for Mellin convolution operators, Mediterr. J. Math., 7 (2010), 483-501.  doi: 10.1007/s00009-010-0062-z.  Google Scholar

[15]

C. Bardaro and I. Mantellini, Multivariate moment type operators: Approximation properties in Orlicz spaces, J. Math. Inequal., 2 (2008), 247-259.  doi: 10.7153/jmi-02-22.  Google Scholar

[16]

C. Bardaro, I. Mantellini, G. Uysal and B. Yılmaz, A class of integral operators that fix exponential functions, Mediterr. J. Math., 18 (2021), Paper No. 179, 21 pp. doi: 10.1007/s00009-021-01819-0.  Google Scholar

[17]

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications 9., Walter De Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110199277.  Google Scholar

[18]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2 (1952), 43-56.  doi: 10.1007/BF02591381.  Google Scholar

[19]

B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 14 (1970), 9-13.   Google Scholar

[20] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation Vol. 1: One-Dimensional Theory, Pure and Applied Mathematics, Vol. 40. Academic Press, New York-London, 1971.  doi: 10.1007/978-3-0348-7448-9.  Google Scholar
[21]

P. L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), 325-376.  doi: 10.1007/BF02649101.  Google Scholar

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183. doi: 10.1007/978-1-4613-9757-1_5.  Google Scholar

[23]

D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34 (2013), 819-844.  doi: 10.1080/01630563.2013.767833.  Google Scholar

[24]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Mathematical Foundations of Computing, 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.  Google Scholar

[25]

A. D. Gadžiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, (Russian), Dokl. Akad. Nauk SSSR, 218 (1974), 1001-1004.   Google Scholar

[26]

V. Gupta and V. K. Singh, Modified Post-Widder operators preserving exponential functions, Advances in Mathematical Methods and High Performance Computing, 41 (2019), 181-192.  doi: 10.1007/978-3-030-02487-1_10.  Google Scholar

[27]

V. Gupta and G. Tachev, On approximation properties of Phillips operators preserving exponential functions, Mediterr. J. Math., 14 (2017), Paper No. 177, 12 pp. doi: 10.1007/s00009-017-0981-z.  Google Scholar

[28]

A. Holhoş, The rate of approximation of functions in an infinite interval by positive linear operators, Stud. Univ. Babeş–Bolyai Math., 55 (2010), 133–142.  Google Scholar

[29]

A. Holhoş, Quantitative estimates of Voronovskaya type in weighted spaces, Results Math., 73 (2018), Paper No. 53, 11 pp. doi: 10.1007/s00025-018-0814-9.  Google Scholar

[30]

H. Karslı, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal., 85 (2006), 781-791.  doi: 10.1080/00036810600712665.  Google Scholar

[31]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.  Google Scholar

[32]

P. P. Korovkin, On convergence of linear positive operators in the spaces of continuous functions, (Russian), Doklady Akad. Nauk. SSSR (N.S.), 90 (1953), 961-964.   Google Scholar

[33]

P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corp., Delhi, 1960.  Google Scholar

[34]

A. Lupaş and M. Müller, Approximationseigenschaften der Gammaoperatoren, (German), Math. Z., 98 (1967), 208-226.  doi: 10.1007/BF01112415.  Google Scholar

[35]

R. G. Mamedov, The Mellin Transform and Approximation Theory, (Russian) "Elm", Baku, 1991.  Google Scholar

[36]

C. P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canadian J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8.  Google Scholar

[37]

I. P. Natanson, Theory of Functions of a Real Variable Vol. Ⅱ., Frederick Ungar Pub. Co., New York, 1961.  Google Scholar

[38]

R. S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. of Math., 59 (1954), 325-356.  doi: 10.2307/1969697.  Google Scholar

[39]

L. Rempulska and K. Tomczak, On some properties of the Picard operators, Arch. Math. (Brno), 45 (2009), 125-135.   Google Scholar

[40] L. L. Schumaker, Spline Functions: Basic Theory, 3$^rd$ edition, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618994.  Google Scholar
[41]

T. Świderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. (Prace Mat.), 40 (2000), 181-189.   Google Scholar

[42]

E. V. Voronovskaya, Determination of the asymptotic form of approximation of functions by the polynomials of S. N. Bernstein, Dokl. Akad. Nauk SSSR, A, (1932), 79–85. Google Scholar

[43]

E. Wachnicki and G. Krech, Approximation of functions by nonlinear singular integral operators depending on two parameters, Publ. Math. Debrecen, 92 (2018), 481-494.  doi: 10.5486/PMD.2018.8080.  Google Scholar

[44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series, Vol. 6. Princeton Univ. Press, Princeton, 1941.   Google Scholar
[45]

B. YılmazG. Uysal and A. Aral, Reconstruction of two approximation processes in order to reproduce $e^ax$ and $e^2ax$, $a>0$, J. Math. Inequal., 15 (2021), 1101-1118.  doi: 10.7153/jmi-2021-15-75.  Google Scholar

Figure 1.  $ a = \frac{1}{2} $ and $ n = 5 $
Figure 2.  $ a = \frac{3}{4} $ and $ n = 5 $
Figure 3.  $ a = \frac{3}{10} $ and $ n = 5 $
Figure 4.  $ a = \frac{1}{2} $ and $ n = 5 $
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