doi: 10.3934/mfc.2022019
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Convergence on sequences of Szász-Jakimovski-Leviatan type operators and related results

Computational & Analytical Mathematics and Their Applications Research Group, Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia

*Corresponding author: Mohammad Nasiruzzaman

Received  February 2022 Revised  May 2022 Early access July 2022

In the present article, we construct the Szász-Jakimovski-Leviatan operators in parametric form by including the sequences of continuous functions and then investigate the approximation properties. We have successfully estimated the convergence by use of modulus of continuity in the spaces of Lipschitz functions, Peetres $ K $-functional and weighted functions.

Citation: Mohammad Nasiruzzaman. Convergence on sequences of Szász-Jakimovski-Leviatan type operators and related results. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022019
References:
[1]

P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Sup., 9 (1880), 119–144. doi: 10.24033/asens.186.

[2]

T. AcarM. Mursaleen and S. A. Mohiuddine, Stancu type $(p, q)$-Szász-Mirakyan-Baskakov operators, Commun. Fac. Sci. Univ. Ank. Series A1., 67 (2018), 116-128.  doi: 10.1501/Commua1_0000000835.

[3]

A. Alotaibi and M. Mursaleen, Approximation of Jakimovski-Leviatan-Beta type integral operators via $q$-calculus, AIMS Mathematics, 5 (2020), 3019-3034.  doi: 10.3934/math.2020196.

[4]

Ç. Atakut and İ. Büyükyazıcı, Approximation by modified integral type Jakimovski-Leviatan operators, Filomat, 30 (2016), 29-39.  doi: 10.2298/FIL1601029C.

[5]

A. Alotaibi, M. Nasiruzzaman and M. Mursaleen, A Dunkl type generalization of Szász operators via post-quantum calculus, J. Inequal. Appl., 2018 (2018), 287, 15 pp. doi: 10.1186/s13660-018-1878-5.

[6]

A. AlotaibiM. Nasiruzzaman and M. Mursaleen, Approximation by Phillips operators via q-Dunkl generalization based on a new parameter, Journal of King Saud University-Science, 33 (2021), 101413.  doi: 10.1016/j.jksus.2021.101413.

[7]

T. AcarA. Aral and S. A. Mohiuddine, Approximation By Bivariate $(p, q)$-Bernstein Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 655-662.  doi: 10.1007/s40995-016-0045-4.

[8]

T. AcarS. A. Mohiuddine and M. Mursaleen, Approximation by $(p, q)$-Baskakov-Durrmeyer-Stancu operators, Comp. Anal. Op. Theory., 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.

[9]

Ç. Atakut and N. İspir, Approximation by modified Szá sz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 571-578.  doi: 10.1007/BF02829690.

[10]

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

[11]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Mathematical Foundations of Computing, 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[12]

Z.-C. GuoD.-H. XiangX. Guo and D.-X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[13]

A. D. Gadžiev, Theorems of the Type of P.P. Korovkin's Theorems, Math. Notes., 20 (1976), 995-998. 

[14]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[15]

A. KajlaS. A. Mohiuddine and A. Alotaibi, Blending-type approximation by Lupaş- Durrmeyer-type operators involving Pólya distribution, Math. Meth. Appl. Sci., 44 (2021), 9407-9418.  doi: 10.1002/mma.7368.

[16]

A. KajlaS. A. Mohiuddine and A. Alotaibi, Durrmeyer-type generalization of $\mu$- Bernstein operators, Filomat, 36 (2022), 349-360.  doi: 10.2298/FIL2201349K.

[17]

P. P. Korovkin, Linear Operators and Approximation Theory, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III Gordon and Breach Publishers, Inc., New York; Hindustan Publishing Corp. (India), Delhi, 1960.

[18]

B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl. Akad. Indag. Math., 50 (1988), 53-63. 

[19]

G. V. MilovanovicM. Mursaleen and M. Nasiruzzaman, Modified Stancu type Dunkl generalization of Szász-Kantorovich operators, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 112 (2018), 135-151.  doi: 10.1007/s13398-016-0369-0.

[20]

M. Mursaleen, M. Nasiruzzaman and A. Alotaibi, On Modified Dunkl generalization of Szász operators via $q$-calculus, J. Inequal. Appl., 2017 (2017), 38, 12 pp. doi: 10.1186/s13660-017-1311-5.

[21]

M. MursaleenK. J. Ansari and A. Khan, Approximation by Kantorovich type $q$-Bernstein-Stancu operators, Complex Anal. Oper. Theory., 11 (2017), 85-107.  doi: 10.1007/s11785-016-0572-1.

[22]

S. A. MohiuddineB. Hazarika and M. A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33 (2019), 4549-4560.  doi: 10.2298/FIL1914549M.

[23]

S. A. MohiuddineA. AhmadF. ÖzgerA. Alotaibi and B. Hazarika, Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran. J. Sci. Technol. Trans. Sci., 45 (2021), 593-605.  doi: 10.1007/s40995-020-01024-w.

[24]

S. A. Mohiuddine and U. Kadak, Generalized statistically almost convergence based on the difference operator which includes the $(p, q)$-gamma function and related approximation theorems, Results Math., 73 (2018), Paper No. 9, 31 pp. doi: 10.1007/s00025-018-0789-6.

[25]

S. A. Mohiuddine, Approximation by bivariate generalized BernsteinSchurer operators and associated GBS operators, Adv. Difference Equ., 2020 (2020), Paper No. 676, 17 pp. doi: 10.1186/s13662-020-03125-7.

[26]

S. A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter $\alpha$, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 114 (2020), 70, 17 pp. doi: 10.1007/s13398-020-00802-w.

[27]

S. A. MohiuddineT. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559.

[28]

M. Nasiruzzaman and A. F. Aljohani, Approximation by Szász-Jakimovski-Leviatan type operators via aid of Appell polynomials, Journal of Functions Spaces, 2020 (2020), 9657489, 11 pp. doi: 10.1155/2020/9657489.

[29]

M. Nasiruzzaman and A. F. Aljohani, Approximation by parametric extension of Szasz-Mirakjan-Kantorovich operators involving the Appell polynomials, Journal of functions spaces, 2020 (2020), 8863664, 11 pp. doi: 10.1155/2020/8863664.

[30]

M. NasiruzzamanK. J. Ansari and M. Mursaleen, On the Parametric Approximation Results of Phillips Operators Involving the q-Appell Polynomials, Iran J. Sci. Technol Trans A Sci., 46 (2022), 251-263.  doi: 10.1007/s40995-021-01219-9.

[31]

F. Özger, On new Bézier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376-393.  doi: 10.31801/cfsuasmas.510382.

[32]

F. Özger, H. M. Srivastava and S. A. Mohiuddine, Approximation of functions by a new class of generalized Bernstein-Schurer operators, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 114 (2020), 173, 21 pp. doi: 10.1007/s13398-020-00903-6.

[33]

B. Wood, Generalized Szász operators for the approximation in the complex domain, SIAM J. Appl. Math., 17 (1969), 790-801.  doi: 10.1137/0117071.

[34]

O. Szász, Generalization of Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Stds., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.

[35]

M. A. Ozarslan and H. Aktuğlu, Local approximation for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.

[36]

N. RaoA. Wafi and A. M. Acu, $q$ -Szász-Durrmeyer type operators based on Dunkl analogue, Complex Anal. Oper. Theory., 13 (2019), 915-934.  doi: 10.1007/s11785-018-0816-3.

[37]

N. RaoM. NasiruzzamanM. Heshamuddin and M. Shadab, Approximation properties by modified Baskakov Durrmeyer operators based on shape parameter-$\alpha$, Iran. J. Sci. Technol. Trans. A Sci., 45 (2021), 1457-1465.  doi: 10.1007/s40995-021-01125-0.

[38]

M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl., 73 (1994), 369-396. 

[39]

S. Sucu, Dunkl analogue of Sz$\acute{a}$sz operators, Appl. Math. Comput., 244 (2014), 42-48.  doi: 10.1016/j.amc.2014.06.088.

[40]

O. Shisha and B. Bond, The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60 (1968), 1196-1200.  doi: 10.1073/pnas.60.4.1196.

[41]

D.X. Zhou, Deep distributed convolutional neural networks, Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

show all references

References:
[1]

P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Sup., 9 (1880), 119–144. doi: 10.24033/asens.186.

[2]

T. AcarM. Mursaleen and S. A. Mohiuddine, Stancu type $(p, q)$-Szász-Mirakyan-Baskakov operators, Commun. Fac. Sci. Univ. Ank. Series A1., 67 (2018), 116-128.  doi: 10.1501/Commua1_0000000835.

[3]

A. Alotaibi and M. Mursaleen, Approximation of Jakimovski-Leviatan-Beta type integral operators via $q$-calculus, AIMS Mathematics, 5 (2020), 3019-3034.  doi: 10.3934/math.2020196.

[4]

Ç. Atakut and İ. Büyükyazıcı, Approximation by modified integral type Jakimovski-Leviatan operators, Filomat, 30 (2016), 29-39.  doi: 10.2298/FIL1601029C.

[5]

A. Alotaibi, M. Nasiruzzaman and M. Mursaleen, A Dunkl type generalization of Szász operators via post-quantum calculus, J. Inequal. Appl., 2018 (2018), 287, 15 pp. doi: 10.1186/s13660-018-1878-5.

[6]

A. AlotaibiM. Nasiruzzaman and M. Mursaleen, Approximation by Phillips operators via q-Dunkl generalization based on a new parameter, Journal of King Saud University-Science, 33 (2021), 101413.  doi: 10.1016/j.jksus.2021.101413.

[7]

T. AcarA. Aral and S. A. Mohiuddine, Approximation By Bivariate $(p, q)$-Bernstein Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 655-662.  doi: 10.1007/s40995-016-0045-4.

[8]

T. AcarS. A. Mohiuddine and M. Mursaleen, Approximation by $(p, q)$-Baskakov-Durrmeyer-Stancu operators, Comp. Anal. Op. Theory., 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.

[9]

Ç. Atakut and N. İspir, Approximation by modified Szá sz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 571-578.  doi: 10.1007/BF02829690.

[10]

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

[11]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Mathematical Foundations of Computing, 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[12]

Z.-C. GuoD.-H. XiangX. Guo and D.-X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[13]

A. D. Gadžiev, Theorems of the Type of P.P. Korovkin's Theorems, Math. Notes., 20 (1976), 995-998. 

[14]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[15]

A. KajlaS. A. Mohiuddine and A. Alotaibi, Blending-type approximation by Lupaş- Durrmeyer-type operators involving Pólya distribution, Math. Meth. Appl. Sci., 44 (2021), 9407-9418.  doi: 10.1002/mma.7368.

[16]

A. KajlaS. A. Mohiuddine and A. Alotaibi, Durrmeyer-type generalization of $\mu$- Bernstein operators, Filomat, 36 (2022), 349-360.  doi: 10.2298/FIL2201349K.

[17]

P. P. Korovkin, Linear Operators and Approximation Theory, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III Gordon and Breach Publishers, Inc., New York; Hindustan Publishing Corp. (India), Delhi, 1960.

[18]

B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl. Akad. Indag. Math., 50 (1988), 53-63. 

[19]

G. V. MilovanovicM. Mursaleen and M. Nasiruzzaman, Modified Stancu type Dunkl generalization of Szász-Kantorovich operators, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 112 (2018), 135-151.  doi: 10.1007/s13398-016-0369-0.

[20]

M. Mursaleen, M. Nasiruzzaman and A. Alotaibi, On Modified Dunkl generalization of Szász operators via $q$-calculus, J. Inequal. Appl., 2017 (2017), 38, 12 pp. doi: 10.1186/s13660-017-1311-5.

[21]

M. MursaleenK. J. Ansari and A. Khan, Approximation by Kantorovich type $q$-Bernstein-Stancu operators, Complex Anal. Oper. Theory., 11 (2017), 85-107.  doi: 10.1007/s11785-016-0572-1.

[22]

S. A. MohiuddineB. Hazarika and M. A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33 (2019), 4549-4560.  doi: 10.2298/FIL1914549M.

[23]

S. A. MohiuddineA. AhmadF. ÖzgerA. Alotaibi and B. Hazarika, Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran. J. Sci. Technol. Trans. Sci., 45 (2021), 593-605.  doi: 10.1007/s40995-020-01024-w.

[24]

S. A. Mohiuddine and U. Kadak, Generalized statistically almost convergence based on the difference operator which includes the $(p, q)$-gamma function and related approximation theorems, Results Math., 73 (2018), Paper No. 9, 31 pp. doi: 10.1007/s00025-018-0789-6.

[25]

S. A. Mohiuddine, Approximation by bivariate generalized BernsteinSchurer operators and associated GBS operators, Adv. Difference Equ., 2020 (2020), Paper No. 676, 17 pp. doi: 10.1186/s13662-020-03125-7.

[26]

S. A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter $\alpha$, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 114 (2020), 70, 17 pp. doi: 10.1007/s13398-020-00802-w.

[27]

S. A. MohiuddineT. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559.

[28]

M. Nasiruzzaman and A. F. Aljohani, Approximation by Szász-Jakimovski-Leviatan type operators via aid of Appell polynomials, Journal of Functions Spaces, 2020 (2020), 9657489, 11 pp. doi: 10.1155/2020/9657489.

[29]

M. Nasiruzzaman and A. F. Aljohani, Approximation by parametric extension of Szasz-Mirakjan-Kantorovich operators involving the Appell polynomials, Journal of functions spaces, 2020 (2020), 8863664, 11 pp. doi: 10.1155/2020/8863664.

[30]

M. NasiruzzamanK. J. Ansari and M. Mursaleen, On the Parametric Approximation Results of Phillips Operators Involving the q-Appell Polynomials, Iran J. Sci. Technol Trans A Sci., 46 (2022), 251-263.  doi: 10.1007/s40995-021-01219-9.

[31]

F. Özger, On new Bézier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376-393.  doi: 10.31801/cfsuasmas.510382.

[32]

F. Özger, H. M. Srivastava and S. A. Mohiuddine, Approximation of functions by a new class of generalized Bernstein-Schurer operators, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM, 114 (2020), 173, 21 pp. doi: 10.1007/s13398-020-00903-6.

[33]

B. Wood, Generalized Szász operators for the approximation in the complex domain, SIAM J. Appl. Math., 17 (1969), 790-801.  doi: 10.1137/0117071.

[34]

O. Szász, Generalization of Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Stds., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.

[35]

M. A. Ozarslan and H. Aktuğlu, Local approximation for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.

[36]

N. RaoA. Wafi and A. M. Acu, $q$ -Szász-Durrmeyer type operators based on Dunkl analogue, Complex Anal. Oper. Theory., 13 (2019), 915-934.  doi: 10.1007/s11785-018-0816-3.

[37]

N. RaoM. NasiruzzamanM. Heshamuddin and M. Shadab, Approximation properties by modified Baskakov Durrmeyer operators based on shape parameter-$\alpha$, Iran. J. Sci. Technol. Trans. A Sci., 45 (2021), 1457-1465.  doi: 10.1007/s40995-021-01125-0.

[38]

M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl., 73 (1994), 369-396. 

[39]

S. Sucu, Dunkl analogue of Sz$\acute{a}$sz operators, Appl. Math. Comput., 244 (2014), 42-48.  doi: 10.1016/j.amc.2014.06.088.

[40]

O. Shisha and B. Bond, The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60 (1968), 1196-1200.  doi: 10.1073/pnas.60.4.1196.

[41]

D.X. Zhou, Deep distributed convolutional neural networks, Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

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