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November  2021, 4(4): 271-280. doi: 10.3934/mfc.2021015

On multidimensional Urysohn type generalized sampling operators

Bolu Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14030, Golkoy-Bolu, Turkey

* Corresponding author: Harun Karsli

Received  May 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

The concern of this study is to construction of a multidimensional version of Urysohn type generalized sampling operators, whose one dimensional case defined and investigated by the author in [28] and [27]. In details, as a continuation of the studies of the author, the paper centers around to investigation of some approximation and asymptotic properties of the aforementioned linear multidimensional Urysohn type generalized sampling operators.

Citation: Harun Karsli. On multidimensional Urysohn type generalized sampling operators. Mathematical Foundations of Computing, 2021, 4 (4) : 271-280. doi: 10.3934/mfc.2021015
References:
[1]

T. AcarD. Costarelli and G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series, Banach J. Math. Anal., 14 (2020), 1481-1508.  doi: 10.1007/s43037-020-00071-0.

[2]

L. AngeloniN. CetinD. CostarelliA. R. Sambucini and G. Vinti, Multivariate sampling Kantorovich operators: Quantitative estimates in Orlicz spaces, Const. Math. Anal., 4 (2021), 229-241.  doi: 10.33205/cma.876890.

[3]

L. AngeloniD. CostarelliM. SeraciniG. Vinti and L. Zampogni, Variation diminishing-type properties for multivariate sampling Kantorovich operators, Boll. Unione Mat. Ital., 13 (2020), 595-605.  doi: 10.1007/s40574-020-00256-3.

[4]

L. AngeloniD. Costarelli and G. Vinti, Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing, Ann. Acad. Sci. Fenn. Math., 45 (2020), 751-770.  doi: 10.5186/aasfm.2020.4532.

[5]

L. Angeloni and G. Vinti, Estimates in variation for multivariate sampling-type operators, Dolomites Research Notes on App., 14 (2021), 1-9. 

[6]

C. BardaroL. Faina and I. Mantellini, Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nachr., 289 (2016), 1702-1720.  doi: 10.1002/mana.201500225.

[7]

C. Bardaro and I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen J. Approx., 6 (2014), 143-165. 

[8]

C. Bardaro and I. Mantellini, On pointwise approximation properties of multivariate semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.

[9]

C. Bardaro and I. Mantellini, A note on the Voronovskaja theorem for Mellin-Fejer convolution operators, Appl. Math. Lett., 24 (2011), 2064-2067.  doi: 10.1016/j.aml.2011.05.043.

[10]

C. Bardaro, I. Mantellini, R. Stens, J. Vautz and G. Vinti, Generalized sampling approximation for multivariate discontinuous signals and application to image processing, New Perspectives on Approximation and Sampling Theory, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2014), 87–114.

[11]

P. L. ButzerA. Fischer and R. L. Stens, Generalized sampling approximation of multivariate signals; theory and some applications, Note Mat. 10, Suppl., 10 (1990), 173-191. 

[12]

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, V.1, Academic Press, New York, London, 40 (1971).

[13]

P. L. ButzerS. Ries and R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39.  doi: 10.1016/0021-9045(87)90063-3.

[14]

P. L. Butzer and R. L. Stens, Sampling theory for not necessarily band-limited functions: A historical overview, SIAM Rev., 34 (1992), 40-53.  doi: 10.1137/1034002.

[15]

D. Costarelli, Neural network operators: Constructive interpolation of multivariate functions, Neural Networks, 67 (2015), 28-36.  doi: 10.1016/j.neunet.2015.02.002.

[16]

D. Costarelli, M. Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comp., 374 (2020), 125046. doi: 10.1016/j.amc.2020.125046.

[17]

D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integral Equations Appl., 26 (2014), 455-481.  doi: 10.1216/JIE-2014-26-4-455.

[18]

D. Costarelli and G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. Unione Mat. Ital., 4 (2011), 445-468. 

[19]

D. Costarelli and G. Vinti, Approximation results by multivariate sampling Kantorovich series in Musielak-Orlicz spaces, Dolomites Res. Notes Approx., 12 (2019), 7-16. 

[20]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Math Found. Comp., 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.

[21]

I. I. Demkiv, On approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L'viv. Univ., Ser. Prykl. Mat. Inform., 2 (2000), 26-30. 

[22]

H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators, Results Math., 72 (2017), 1571-1583.  doi: 10.1007/s00025-017-0729-x.

[23]

H. Karsli, Approximation results for Urysohn type nonlinear Bernstein operators, Advances in Summability and Approximation Theory, Springer, Singapore, (2018), 223–241.

[24]

H. Karsli, Approximation results for Urysohn type two dimensional nonlinear Bernstein operators, Const. Math. Anal., 1 (2018), 45-57.  doi: 10.33205/cma.453027.

[25]

H. Karsli, Voronovskaya-type theorems for Urysohn type nonlinear Bernstein operators, Math. Methods Appl. Sci., 42 (2019), 5190-5198.  doi: 10.1002/mma.5261.

[26]

H. Karsli, Some approximation properties of urysohn type nonlinear operators, Stud. Univ. Babeş-Bolyai Math., 64 (2019), 183–196. doi: 10.24193/subbmath.2019.2.05.

[27]

H. Karsli, On urysohn type generalized sampling operators, Dolomites Research Notes on Approximation, 14 (2021), 58-67. 

[28]

H. Karsli, Asymptotic properties of urysohn type generalized sampling operators, Carpathian Math. Publ., (accepted), (2021).

[29]

V. L. Makarov and I. I. Demkiv, Approximation of the Urysohn operator by operator polynomials of Stancu type, Ukrainian Math. J., 64 (2012), 356-386.  doi: 10.1007/s11253-012-0652-y.

[30]

S. Ries and R. L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions (Bl. Sendov, P. Petrushev, R. Maalev, and S. Tashev, eds.). Pugl. House Bulgarian Academy of Sciences, Sofia, (1984), 746–756.

show all references

References:
[1]

T. AcarD. Costarelli and G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series, Banach J. Math. Anal., 14 (2020), 1481-1508.  doi: 10.1007/s43037-020-00071-0.

[2]

L. AngeloniN. CetinD. CostarelliA. R. Sambucini and G. Vinti, Multivariate sampling Kantorovich operators: Quantitative estimates in Orlicz spaces, Const. Math. Anal., 4 (2021), 229-241.  doi: 10.33205/cma.876890.

[3]

L. AngeloniD. CostarelliM. SeraciniG. Vinti and L. Zampogni, Variation diminishing-type properties for multivariate sampling Kantorovich operators, Boll. Unione Mat. Ital., 13 (2020), 595-605.  doi: 10.1007/s40574-020-00256-3.

[4]

L. AngeloniD. Costarelli and G. Vinti, Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing, Ann. Acad. Sci. Fenn. Math., 45 (2020), 751-770.  doi: 10.5186/aasfm.2020.4532.

[5]

L. Angeloni and G. Vinti, Estimates in variation for multivariate sampling-type operators, Dolomites Research Notes on App., 14 (2021), 1-9. 

[6]

C. BardaroL. Faina and I. Mantellini, Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nachr., 289 (2016), 1702-1720.  doi: 10.1002/mana.201500225.

[7]

C. Bardaro and I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen J. Approx., 6 (2014), 143-165. 

[8]

C. Bardaro and I. Mantellini, On pointwise approximation properties of multivariate semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.

[9]

C. Bardaro and I. Mantellini, A note on the Voronovskaja theorem for Mellin-Fejer convolution operators, Appl. Math. Lett., 24 (2011), 2064-2067.  doi: 10.1016/j.aml.2011.05.043.

[10]

C. Bardaro, I. Mantellini, R. Stens, J. Vautz and G. Vinti, Generalized sampling approximation for multivariate discontinuous signals and application to image processing, New Perspectives on Approximation and Sampling Theory, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2014), 87–114.

[11]

P. L. ButzerA. Fischer and R. L. Stens, Generalized sampling approximation of multivariate signals; theory and some applications, Note Mat. 10, Suppl., 10 (1990), 173-191. 

[12]

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, V.1, Academic Press, New York, London, 40 (1971).

[13]

P. L. ButzerS. Ries and R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39.  doi: 10.1016/0021-9045(87)90063-3.

[14]

P. L. Butzer and R. L. Stens, Sampling theory for not necessarily band-limited functions: A historical overview, SIAM Rev., 34 (1992), 40-53.  doi: 10.1137/1034002.

[15]

D. Costarelli, Neural network operators: Constructive interpolation of multivariate functions, Neural Networks, 67 (2015), 28-36.  doi: 10.1016/j.neunet.2015.02.002.

[16]

D. Costarelli, M. Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comp., 374 (2020), 125046. doi: 10.1016/j.amc.2020.125046.

[17]

D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integral Equations Appl., 26 (2014), 455-481.  doi: 10.1216/JIE-2014-26-4-455.

[18]

D. Costarelli and G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. Unione Mat. Ital., 4 (2011), 445-468. 

[19]

D. Costarelli and G. Vinti, Approximation results by multivariate sampling Kantorovich series in Musielak-Orlicz spaces, Dolomites Res. Notes Approx., 12 (2019), 7-16. 

[20]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Math Found. Comp., 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.

[21]

I. I. Demkiv, On approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L'viv. Univ., Ser. Prykl. Mat. Inform., 2 (2000), 26-30. 

[22]

H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators, Results Math., 72 (2017), 1571-1583.  doi: 10.1007/s00025-017-0729-x.

[23]

H. Karsli, Approximation results for Urysohn type nonlinear Bernstein operators, Advances in Summability and Approximation Theory, Springer, Singapore, (2018), 223–241.

[24]

H. Karsli, Approximation results for Urysohn type two dimensional nonlinear Bernstein operators, Const. Math. Anal., 1 (2018), 45-57.  doi: 10.33205/cma.453027.

[25]

H. Karsli, Voronovskaya-type theorems for Urysohn type nonlinear Bernstein operators, Math. Methods Appl. Sci., 42 (2019), 5190-5198.  doi: 10.1002/mma.5261.

[26]

H. Karsli, Some approximation properties of urysohn type nonlinear operators, Stud. Univ. Babeş-Bolyai Math., 64 (2019), 183–196. doi: 10.24193/subbmath.2019.2.05.

[27]

H. Karsli, On urysohn type generalized sampling operators, Dolomites Research Notes on Approximation, 14 (2021), 58-67. 

[28]

H. Karsli, Asymptotic properties of urysohn type generalized sampling operators, Carpathian Math. Publ., (accepted), (2021).

[29]

V. L. Makarov and I. I. Demkiv, Approximation of the Urysohn operator by operator polynomials of Stancu type, Ukrainian Math. J., 64 (2012), 356-386.  doi: 10.1007/s11253-012-0652-y.

[30]

S. Ries and R. L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions (Bl. Sendov, P. Petrushev, R. Maalev, and S. Tashev, eds.). Pugl. House Bulgarian Academy of Sciences, Sofia, (1984), 746–756.

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