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A Note On Kantorovich Type Operators Which Preserve Affine Functions

Year 2024, Volume: 7 Issue: 1, 53 - 58, 31.03.2024
https://doi.org/10.33401/fujma.1424382

Abstract

The authors present an integral widening of operators which preserve affine functions. Influenced by the operators which preserve affine functions, we define the integral extension of these operators. We give quantitative type theorem using weighted modulus of continuity. Withal quantitative Voronovskaya theorem is aquired by classical modulus of continuity. When the moments of the operator are known, convergence results with the moments obtained for the Kantorovich form of the same operator is given.

References

  • [1] A. Aral, D. Cardenas-Morales, P. Garrancho and I. Ras¸a, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62(1) (2011), 158-163. $\href{https://doi.org/10.1016/j.camwa.2011.04.063}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-79959510093&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bernstein-type+operators+which+preserve+polynomials%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000292853300016}{[\mbox{Web of Science}]} $
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  • [3] A. Aral, D. Aydın Arı and B. Yılmaz, A Note On Kantorovich Type Bernstein Chlodovsky Operator Which Preserve Exponential Functions, J. Math. Inequal., 15(3), (2021), 1173-1183. $\href{https://doi.org/10.7153/jmi-2021-15-78}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85116803023&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+Kantorovich+type+Bernstein+Chlodovsky+operators+which+preserve+exponential+function%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000705523600017}{[\mbox{Web of Science}]}$
  • [4] A. Aral, D. Otrocol and I. Raşa, On approximation by some Bernstein–Kantorovich exponential-type polynomials, Period. Math. Hung.,79 (2) (2019), 236-253. $\href{https://doi.org/10.1007/s10998-019-00284-3}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85068311018&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+approximation+by+some+Bernstein%E2%80%93Kantorovich+exponential-type+polynomials%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000492157300011}{[\mbox{Web of Science}]} $
  • [5] K.J. Ansari, On Kantorovich variant of Baskakov type operators preserving some functions, Filomat, 36(3) (2022), 1049–1060. $\href{https://doi.org/10.2298/FIL2203049A}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85127436273&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+Kantorovich+variant+of+Baskakov+type+operators+preserving+some+functions%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000778010200026}{[\mbox{Web of Science}]}$
  • [6] K.J. Ansari, S. Karakılıç and F. Özger, Bivariate Bernstein-Kantorovich operators with a summability method and related GBS operators, Filomat, 36(19), (2022), 6751-6765. $\href{https://doi.org/10.2298/FIL2219751A}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85146789845&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bivariate+Bernstein-Kantorovich+operators+with+a+summability+method+and+related+GBS+operators%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000945598000022}{[\mbox{Web of Science}]}$
  • [7] S. Rahman and K.J. Ansari, Estimation using a summation integral operator of exponential type with a weight derived from the a-Baskakov basis function, Math. Methods Appl. Sci., 47(4), (2024), 2535-2547. $\href{https://doi.org/10.1002/mma.9763}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85175971378&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Estimation+using+a+summation+integral+operator+of+exponential+type+with+a+weight+derived+from+the%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001096921900001}{[\mbox{Web of Science}]}$
  • [8] F. Usta, M. Akyiğit, F. Say and K.J. Ansari, Bernstein operator method for approximate solution of singularly perturbed Volterra integral equations, Journal of Mathematical Analysis and Applications, 507(2), (2022) 125828. $\href{https://doi.org/10.1016/j.jmaa.2021.125828}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85119321719&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bernstein+operator+method+for+approximate+solution+of+singularly+perturbed+Volterra+integral+equations%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000775539700026}{[\mbox{Web of Science}]} $
  • [9] L.V. Kantorovich, Sur certains d’eveloppementssuivant les polynmes de la forme de S. Bernstein, I,II.C.R. Acad.URSS, (1930), 563-568 and 595-600.
  • [10] A.D. Gadjiev, On P.P. Korovkin type theorems., Math. Zametki, 20(5) (1976), 995-998. $\href{https://doi.org/10.1007/BF01146928}{[\mbox{CrossRef}]}$
  • [11] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Engl. Translated. Sov.Math. Dokl., 15 (1974), 1433-1436.
  • [12] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and Its Applications, Walter de Gruyter, New York, (1994). $\href{https://doi.org/10.1515/9783110884586}{[\mbox{CrossRef}]}$
  • [13] N. Ispir, On modified Baskakov operators on weighted spaces, Turk. J. Math., 25(3) (2001), 355-365. $\href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85006247329&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Kantorovich+type+operators+preserving+affine+functions%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]}$
Year 2024, Volume: 7 Issue: 1, 53 - 58, 31.03.2024
https://doi.org/10.33401/fujma.1424382

Abstract

References

  • [1] A. Aral, D. Cardenas-Morales, P. Garrancho and I. Ras¸a, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62(1) (2011), 158-163. $\href{https://doi.org/10.1016/j.camwa.2011.04.063}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-79959510093&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bernstein-type+operators+which+preserve+polynomials%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000292853300016}{[\mbox{Web of Science}]} $
  • [2] O. Agratini, Kantorovich type operators preserving affine functions, Hacettepe J. Math. Stat.,45(6) (2016),1657-1663. $\href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85006247329&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Kantorovich+type+operators+preserving+affine+functions%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000392742200001}{[\mbox{Web of Science}]} $
  • [3] A. Aral, D. Aydın Arı and B. Yılmaz, A Note On Kantorovich Type Bernstein Chlodovsky Operator Which Preserve Exponential Functions, J. Math. Inequal., 15(3), (2021), 1173-1183. $\href{https://doi.org/10.7153/jmi-2021-15-78}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85116803023&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+Kantorovich+type+Bernstein+Chlodovsky+operators+which+preserve+exponential+function%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000705523600017}{[\mbox{Web of Science}]}$
  • [4] A. Aral, D. Otrocol and I. Raşa, On approximation by some Bernstein–Kantorovich exponential-type polynomials, Period. Math. Hung.,79 (2) (2019), 236-253. $\href{https://doi.org/10.1007/s10998-019-00284-3}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85068311018&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+approximation+by+some+Bernstein%E2%80%93Kantorovich+exponential-type+polynomials%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000492157300011}{[\mbox{Web of Science}]} $
  • [5] K.J. Ansari, On Kantorovich variant of Baskakov type operators preserving some functions, Filomat, 36(3) (2022), 1049–1060. $\href{https://doi.org/10.2298/FIL2203049A}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85127436273&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+Kantorovich+variant+of+Baskakov+type+operators+preserving+some+functions%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000778010200026}{[\mbox{Web of Science}]}$
  • [6] K.J. Ansari, S. Karakılıç and F. Özger, Bivariate Bernstein-Kantorovich operators with a summability method and related GBS operators, Filomat, 36(19), (2022), 6751-6765. $\href{https://doi.org/10.2298/FIL2219751A}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85146789845&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bivariate+Bernstein-Kantorovich+operators+with+a+summability+method+and+related+GBS+operators%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000945598000022}{[\mbox{Web of Science}]}$
  • [7] S. Rahman and K.J. Ansari, Estimation using a summation integral operator of exponential type with a weight derived from the a-Baskakov basis function, Math. Methods Appl. Sci., 47(4), (2024), 2535-2547. $\href{https://doi.org/10.1002/mma.9763}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85175971378&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Estimation+using+a+summation+integral+operator+of+exponential+type+with+a+weight+derived+from+the%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001096921900001}{[\mbox{Web of Science}]}$
  • [8] F. Usta, M. Akyiğit, F. Say and K.J. Ansari, Bernstein operator method for approximate solution of singularly perturbed Volterra integral equations, Journal of Mathematical Analysis and Applications, 507(2), (2022) 125828. $\href{https://doi.org/10.1016/j.jmaa.2021.125828}{[\mbox{CrossRef}]} \href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85119321719&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bernstein+operator+method+for+approximate+solution+of+singularly+perturbed+Volterra+integral+equations%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000775539700026}{[\mbox{Web of Science}]} $
  • [9] L.V. Kantorovich, Sur certains d’eveloppementssuivant les polynmes de la forme de S. Bernstein, I,II.C.R. Acad.URSS, (1930), 563-568 and 595-600.
  • [10] A.D. Gadjiev, On P.P. Korovkin type theorems., Math. Zametki, 20(5) (1976), 995-998. $\href{https://doi.org/10.1007/BF01146928}{[\mbox{CrossRef}]}$
  • [11] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Engl. Translated. Sov.Math. Dokl., 15 (1974), 1433-1436.
  • [12] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and Its Applications, Walter de Gruyter, New York, (1994). $\href{https://doi.org/10.1515/9783110884586}{[\mbox{CrossRef}]}$
  • [13] N. Ispir, On modified Baskakov operators on weighted spaces, Turk. J. Math., 25(3) (2001), 355-365. $\href{https://www.\mbox{Scopus}.com/record/display.uri?eid=2-s2.0-85006247329&origin=resultslist&sort=plf-f&src=s&sid=12738d0892cb91a336fec2f7d15edceb&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Kantorovich+type+operators+preserving+affine+functions%22%29&sl=55&sessionSearchId=12738d0892cb91a336fec2f7d15edceb&relpos=0}{[\mbox{Scopus}]}$
There are 13 citations in total.

Details

Primary Language English
Subjects Approximation Theory and Asymptotic Methods
Journal Section Articles
Authors

Didem Aydın Arı 0000-0002-5527-8232

Gizem Uğur Yılmaz 0000-0002-5390-2572

Early Pub Date March 29, 2024
Publication Date March 31, 2024
Submission Date January 23, 2024
Acceptance Date March 11, 2024
Published in Issue Year 2024 Volume: 7 Issue: 1

Cite

APA Aydın Arı, D., & Uğur Yılmaz, G. (2024). A Note On Kantorovich Type Operators Which Preserve Affine Functions. Fundamental Journal of Mathematics and Applications, 7(1), 53-58. https://doi.org/10.33401/fujma.1424382
AMA Aydın Arı D, Uğur Yılmaz G. A Note On Kantorovich Type Operators Which Preserve Affine Functions. Fundam. J. Math. Appl. March 2024;7(1):53-58. doi:10.33401/fujma.1424382
Chicago Aydın Arı, Didem, and Gizem Uğur Yılmaz. “A Note On Kantorovich Type Operators Which Preserve Affine Functions”. Fundamental Journal of Mathematics and Applications 7, no. 1 (March 2024): 53-58. https://doi.org/10.33401/fujma.1424382.
EndNote Aydın Arı D, Uğur Yılmaz G (March 1, 2024) A Note On Kantorovich Type Operators Which Preserve Affine Functions. Fundamental Journal of Mathematics and Applications 7 1 53–58.
IEEE D. Aydın Arı and G. Uğur Yılmaz, “A Note On Kantorovich Type Operators Which Preserve Affine Functions”, Fundam. J. Math. Appl., vol. 7, no. 1, pp. 53–58, 2024, doi: 10.33401/fujma.1424382.
ISNAD Aydın Arı, Didem - Uğur Yılmaz, Gizem. “A Note On Kantorovich Type Operators Which Preserve Affine Functions”. Fundamental Journal of Mathematics and Applications 7/1 (March 2024), 53-58. https://doi.org/10.33401/fujma.1424382.
JAMA Aydın Arı D, Uğur Yılmaz G. A Note On Kantorovich Type Operators Which Preserve Affine Functions. Fundam. J. Math. Appl. 2024;7:53–58.
MLA Aydın Arı, Didem and Gizem Uğur Yılmaz. “A Note On Kantorovich Type Operators Which Preserve Affine Functions”. Fundamental Journal of Mathematics and Applications, vol. 7, no. 1, 2024, pp. 53-58, doi:10.33401/fujma.1424382.
Vancouver Aydın Arı D, Uğur Yılmaz G. A Note On Kantorovich Type Operators Which Preserve Affine Functions. Fundam. J. Math. Appl. 2024;7(1):53-8.

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