doi: 10.3934/mfc.2021037
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Approximation by pseudo-linear discrete operators

1. 

Hacettepe University, Department of Mathematics, Çankaya, TR-06800, Ankara, Turkey

2. 

İstanbul Gedik University, Faculty of Engineering, Department of Computer Engineering, 34876, İstanbul, Turkey

* Corresponding author: İsmail Aslan

Received  July 2021 Revised  October 2021 Early access December 2021

Fund Project: This study is supported financially by the Scientific and Technological Research Council of Turkey (TÜBÏTAK; project number: 119F262), for which we are thankful

In this note, we construct a pseudo-linear kind discrete operator based on the continuous and nondecreasing generator function. Then, we obtain an approximation to uniformly continuous functions through this new operator. Furthermore, we calculate the error estimation of this approach with a modulus of continuity based on a generator function. The obtained results are supported by visualizing with an explicit example. Finally, we investigate the relation between discrete operators and generalized sampling series.

Citation: İsmail Aslan, Türkan Yeliz Gökçer. Approximation by pseudo-linear discrete operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021037
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.  Google Scholar

[2]

L. AngeloniD. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767.  doi: 10.5186/aasfm.2018.4343.  Google Scholar

[3]

L. Angeloni and G. Vinti, Discrete operators of sampling type and approximation in $ \varphi$-variation, Math. Nachr., 291 (2018), 546-555.  doi: 10.1002/mana.201600508.  Google Scholar

[4]

İ. Aslan, Approximation by sampling type discrete operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 969-980.  doi: 10.31801/cfsuasmas.671237.  Google Scholar

[5]

İ. Aslan, Convergence in phi-variation and rate of approximation for nonlinear integral operators using summability process, Mediterr. J. Math., 18 (2021), Paper No. 5, 19 pp. doi: 10.1007/s00009-020-01623-2.  Google Scholar

[6]

İ. Aslan, Approximation by sampling-type nonlinear discrete operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 969-980.  doi: 10.31801/cfsuasmas.671237.  Google Scholar

[7]

I. Aslan and O. Duman, Summability on Mellin-type nonlinear integral operators, Integral Transform. Spec. Funct., 30 (2019), 492-511.  doi: 10.1080/10652469.2019.1594209.  Google Scholar

[8]

I. Aslan and O. Duman, Approximation by nonlinear integral operators via summability process, Math. Nachr., 293 (2020), 430-448.  doi: 10.1002/mana.201800187.  Google Scholar

[9]

I. Aslan and O. Duman, Characterization of absolute and uniform continuity, Hacet. J. Math. Stat., 49 (2020), 1550-1565.  doi: 10.15672/hujms.585581.  Google Scholar

[10]

I. Aslan and O. Duman, Nonlinear approximation in N-dimension with the help of summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115 (2021), Paper No. 105, 27 pp. doi: 10.1007/s13398-021-01046-y.  Google Scholar

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B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, Cham, 2016. doi: 10.1007/978-3-319-34189-7.  Google Scholar

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B. Bede and O'R. Donal, The theory of pseudo-linear operators, Knowledge-Based Systems, 38 (2013), 19-26.  doi: 10.1016/j.knosys.2012.07.003.  Google Scholar

[13]

B. BedeH. NobuharaM. Daňková and A. Di Nola, Approximation by pseudo-linear operators, Fuzzy Sets and Systems, 159 (2008), 804-820.  doi: 10.1016/j.fss.2007.11.007.  Google Scholar

[14]

B. BedeE. D. SchwabH. Nobuhara and I. J. Rudas, Approximation by Shepard type pseudo-linear operators and applications to image processing, Internat. J. Approx. Reason, 50 (2009), 21-36.  doi: 10.1016/j.ijar.2008.01.007.  Google Scholar

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L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend., 12 (1993), 511-534.  doi: 10.4171/ZAA/550.  Google Scholar

[16]

A. Boccuto and X. Dimitriou, Rates of approximation for general sampling-type operators in the setting of filter convergence, Appl. Math. Comput., 229 (2014), 214-226.  doi: 10.1016/j.amc.2013.12.044.  Google Scholar

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P. L. ButzerW. Splettstösser and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70.   Google Scholar

[18]

P. L. Butzer and R. L. Stens, Prediction of non-bandlimited signals from past samples in terms of splines of low degree, Math. Nachr., 132 (1987), 115-130.  doi: 10.1002/mana.19871320109.  Google Scholar

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P. L. Butzer and R. L. Stens, Linear predictions in terms of samples from the past: An overview, Numerical Methods and Approximation Theory Ⅲ, (1988), 1-22.   Google Scholar

[20]

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.  Google Scholar

[21]

C.-C. Chiu and W.-J. Wang, A simple computation of MIN and MAX operations for fuzzy numbers, Fuzzy Sets and Systems, 126 (2002), 273-276.  doi: 10.1016/S0165-0114(01)00041-0.  Google Scholar

[22]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, The max-product generalized sampling operators: Convergence and quantitative estimates, Appl. Math. Comput., 355 (2019), 173-183.  doi: 10.1016/j.amc.2019.02.076.  Google Scholar

[23]

L. Coroianu and S. G. Gal, Saturation and inverse results for the Bernstein max-product operator, Period. Math. Hungar., 69 (2014), 126-133.  doi: 10.1007/s10998-014-0062-z.  Google Scholar

[24]

L. Coroianu and S. G. Gal, Approximation by truncated max-product operators of Kantorovich-type based on generalized $ (\phi,\psi)$-kernels, Math. Methods Appl. Sci., 41 (2018), 7971-7984.  doi: 10.1002/mma.5262.  Google Scholar

[25]

L. CoroianuS. G. Gal and B. Bede, Approximation of fuzzy numbers by max-product Bernstein operators, Fuzzy Sets and Systems, 257 (2014), 41-66.  doi: 10.1016/j.fss.2013.04.010.  Google Scholar

[26]

D. Costarelli and A. R. Sambucini, Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators, Results Math., 73 (2018), Art. 15, 15 pp. doi: 10.1007/s00025-018-0799-4.  Google Scholar

[27]

D. CostarelliA. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications, Neural Comput. Appl., 31 (2019), 5069-5078.  doi: 10.1007/s00521-018-03998-6.  Google Scholar

[28]

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. Comput., 374 (2020), 125046, 18pp. doi: 10.1016/j.amc.2020.125046.  Google Scholar

[29]

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

[30]

D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory, 209 (2016), 1-22.  doi: 10.1016/j.jat.2016.05.001.  Google Scholar

[31]

D. Costarelli and G. Vinti, Estimates for the neural network operators of the max-product type with continuous and p-integrable functions, Results Math., 73 (2018), Art. 12, 10 pp. doi: 10.1007/s00025-018-0790-0.  Google Scholar

[32]

O. Duman, Statistical convergence of max-product approximating operators, Turkish J. Math., 34 (2010), 501-514.   Google Scholar

[33]

T. Y. Gokcer and O. Duman, Summation process by max-product operators, Computational Analysis, Springer Proc. Math. Stat., Springer, New York, 155 (2016), 59–67. doi: 10.1007/978-3-319-28443-9_4.  Google Scholar

[34]

T. Y. Gokcer and O. Duman, Approximation by max-min operators: A general theory and its applications, Fuzzy Sets and Systems, 394 (2020), 146-161.  doi: 10.1016/j.fss.2019.11.007.  Google Scholar

[35]

T. Y. Gokcer and O. Duman, Regular summability methods in the approximation by max-min operators, Fuzzy Sets and Systems, 426 (2022), 106-120.  doi: 10.1016/j.fss.2021.03.003.  Google Scholar

[36]

M. Gondran and M. Minoux, Dioïds and semirings: Links to fuzzy sets and other applications, Fuzzy Sets and Systems, 158 (2007), 1273-1294.  doi: 10.1016/j.fss.2007.01.016.  Google Scholar

[37]

C.-C. LiuY.-K. WuY.-Y. Lur and C.-L. Tsai, On the power sequence of a fuzzy matrix with convex combination of max-product and max-min operations, Fuzzy Sets and Systems, 289 (2016), 157-163.  doi: 10.1016/j.fss.2015.06.010.  Google Scholar

[38]

S. Ries and R. L. Stens, Approximation by generalized sampling series, Proceedings of the International Conference on Constructive Theory of Functions (Varna, 1984), Bulgarian Academy of Science, Sofia, (1984), 746–756. Google Scholar

[39]

R.-H. Shen and L.-Y. Wei, Convexity of functions produced by Bernstein operators of max-product kind, Results Math., 74 (2019), Art. 92, 6 pp. doi: 10.1007/s00025-019-1015-x.  Google Scholar

[40]

D. Shepard, A two-dimensional interpolation function for irregularly spaced data, Proc. 1968 ACM National Conference, (1968), 517-524.  doi: 10.1145/800186.810616.  Google Scholar

[41]

H. TahayoriA. G. B. TettamanziG. Degli Antoni and A. Visconti, On the calculation of extended max and min operations between convex fuzzy sets of the real line, Fuzzy Sets and Systems, 160 (2009), 3103-3114.  doi: 10.1016/j.fss.2009.06.005.  Google Scholar

[42]

L. Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems, 17 (1985), 313-328.  doi: 10.1016/0165-0114(85)90096-X.  Google Scholar

[43]

X. ZhangS. TanC.-C. Hang and P.-Z. Wang, An efficient computational algorithm for min-max operations, Fuzzy Sets and Systems, 104 (1999), 297-304.  doi: 10.1016/S0165-0114(97)00207-8.  Google Scholar

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.  Google Scholar

[2]

L. AngeloniD. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767.  doi: 10.5186/aasfm.2018.4343.  Google Scholar

[3]

L. Angeloni and G. Vinti, Discrete operators of sampling type and approximation in $ \varphi$-variation, Math. Nachr., 291 (2018), 546-555.  doi: 10.1002/mana.201600508.  Google Scholar

[4]

İ. Aslan, Approximation by sampling type discrete operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 969-980.  doi: 10.31801/cfsuasmas.671237.  Google Scholar

[5]

İ. Aslan, Convergence in phi-variation and rate of approximation for nonlinear integral operators using summability process, Mediterr. J. Math., 18 (2021), Paper No. 5, 19 pp. doi: 10.1007/s00009-020-01623-2.  Google Scholar

[6]

İ. Aslan, Approximation by sampling-type nonlinear discrete operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 969-980.  doi: 10.31801/cfsuasmas.671237.  Google Scholar

[7]

I. Aslan and O. Duman, Summability on Mellin-type nonlinear integral operators, Integral Transform. Spec. Funct., 30 (2019), 492-511.  doi: 10.1080/10652469.2019.1594209.  Google Scholar

[8]

I. Aslan and O. Duman, Approximation by nonlinear integral operators via summability process, Math. Nachr., 293 (2020), 430-448.  doi: 10.1002/mana.201800187.  Google Scholar

[9]

I. Aslan and O. Duman, Characterization of absolute and uniform continuity, Hacet. J. Math. Stat., 49 (2020), 1550-1565.  doi: 10.15672/hujms.585581.  Google Scholar

[10]

I. Aslan and O. Duman, Nonlinear approximation in N-dimension with the help of summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115 (2021), Paper No. 105, 27 pp. doi: 10.1007/s13398-021-01046-y.  Google Scholar

[11]

B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, Cham, 2016. doi: 10.1007/978-3-319-34189-7.  Google Scholar

[12]

B. Bede and O'R. Donal, The theory of pseudo-linear operators, Knowledge-Based Systems, 38 (2013), 19-26.  doi: 10.1016/j.knosys.2012.07.003.  Google Scholar

[13]

B. BedeH. NobuharaM. Daňková and A. Di Nola, Approximation by pseudo-linear operators, Fuzzy Sets and Systems, 159 (2008), 804-820.  doi: 10.1016/j.fss.2007.11.007.  Google Scholar

[14]

B. BedeE. D. SchwabH. Nobuhara and I. J. Rudas, Approximation by Shepard type pseudo-linear operators and applications to image processing, Internat. J. Approx. Reason, 50 (2009), 21-36.  doi: 10.1016/j.ijar.2008.01.007.  Google Scholar

[15]

L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend., 12 (1993), 511-534.  doi: 10.4171/ZAA/550.  Google Scholar

[16]

A. Boccuto and X. Dimitriou, Rates of approximation for general sampling-type operators in the setting of filter convergence, Appl. Math. Comput., 229 (2014), 214-226.  doi: 10.1016/j.amc.2013.12.044.  Google Scholar

[17]

P. L. ButzerW. Splettstösser and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70.   Google Scholar

[18]

P. L. Butzer and R. L. Stens, Prediction of non-bandlimited signals from past samples in terms of splines of low degree, Math. Nachr., 132 (1987), 115-130.  doi: 10.1002/mana.19871320109.  Google Scholar

[19]

P. L. Butzer and R. L. Stens, Linear predictions in terms of samples from the past: An overview, Numerical Methods and Approximation Theory Ⅲ, (1988), 1-22.   Google Scholar

[20]

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.  Google Scholar

[21]

C.-C. Chiu and W.-J. Wang, A simple computation of MIN and MAX operations for fuzzy numbers, Fuzzy Sets and Systems, 126 (2002), 273-276.  doi: 10.1016/S0165-0114(01)00041-0.  Google Scholar

[22]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, The max-product generalized sampling operators: Convergence and quantitative estimates, Appl. Math. Comput., 355 (2019), 173-183.  doi: 10.1016/j.amc.2019.02.076.  Google Scholar

[23]

L. Coroianu and S. G. Gal, Saturation and inverse results for the Bernstein max-product operator, Period. Math. Hungar., 69 (2014), 126-133.  doi: 10.1007/s10998-014-0062-z.  Google Scholar

[24]

L. Coroianu and S. G. Gal, Approximation by truncated max-product operators of Kantorovich-type based on generalized $ (\phi,\psi)$-kernels, Math. Methods Appl. Sci., 41 (2018), 7971-7984.  doi: 10.1002/mma.5262.  Google Scholar

[25]

L. CoroianuS. G. Gal and B. Bede, Approximation of fuzzy numbers by max-product Bernstein operators, Fuzzy Sets and Systems, 257 (2014), 41-66.  doi: 10.1016/j.fss.2013.04.010.  Google Scholar

[26]

D. Costarelli and A. R. Sambucini, Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators, Results Math., 73 (2018), Art. 15, 15 pp. doi: 10.1007/s00025-018-0799-4.  Google Scholar

[27]

D. CostarelliA. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications, Neural Comput. Appl., 31 (2019), 5069-5078.  doi: 10.1007/s00521-018-03998-6.  Google Scholar

[28]

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. Comput., 374 (2020), 125046, 18pp. doi: 10.1016/j.amc.2020.125046.  Google Scholar

[29]

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

[30]

D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory, 209 (2016), 1-22.  doi: 10.1016/j.jat.2016.05.001.  Google Scholar

[31]

D. Costarelli and G. Vinti, Estimates for the neural network operators of the max-product type with continuous and p-integrable functions, Results Math., 73 (2018), Art. 12, 10 pp. doi: 10.1007/s00025-018-0790-0.  Google Scholar

[32]

O. Duman, Statistical convergence of max-product approximating operators, Turkish J. Math., 34 (2010), 501-514.   Google Scholar

[33]

T. Y. Gokcer and O. Duman, Summation process by max-product operators, Computational Analysis, Springer Proc. Math. Stat., Springer, New York, 155 (2016), 59–67. doi: 10.1007/978-3-319-28443-9_4.  Google Scholar

[34]

T. Y. Gokcer and O. Duman, Approximation by max-min operators: A general theory and its applications, Fuzzy Sets and Systems, 394 (2020), 146-161.  doi: 10.1016/j.fss.2019.11.007.  Google Scholar

[35]

T. Y. Gokcer and O. Duman, Regular summability methods in the approximation by max-min operators, Fuzzy Sets and Systems, 426 (2022), 106-120.  doi: 10.1016/j.fss.2021.03.003.  Google Scholar

[36]

M. Gondran and M. Minoux, Dioïds and semirings: Links to fuzzy sets and other applications, Fuzzy Sets and Systems, 158 (2007), 1273-1294.  doi: 10.1016/j.fss.2007.01.016.  Google Scholar

[37]

C.-C. LiuY.-K. WuY.-Y. Lur and C.-L. Tsai, On the power sequence of a fuzzy matrix with convex combination of max-product and max-min operations, Fuzzy Sets and Systems, 289 (2016), 157-163.  doi: 10.1016/j.fss.2015.06.010.  Google Scholar

[38]

S. Ries and R. L. Stens, Approximation by generalized sampling series, Proceedings of the International Conference on Constructive Theory of Functions (Varna, 1984), Bulgarian Academy of Science, Sofia, (1984), 746–756. Google Scholar

[39]

R.-H. Shen and L.-Y. Wei, Convexity of functions produced by Bernstein operators of max-product kind, Results Math., 74 (2019), Art. 92, 6 pp. doi: 10.1007/s00025-019-1015-x.  Google Scholar

[40]

D. Shepard, A two-dimensional interpolation function for irregularly spaced data, Proc. 1968 ACM National Conference, (1968), 517-524.  doi: 10.1145/800186.810616.  Google Scholar

[41]

H. TahayoriA. G. B. TettamanziG. Degli Antoni and A. Visconti, On the calculation of extended max and min operations between convex fuzzy sets of the real line, Fuzzy Sets and Systems, 160 (2009), 3103-3114.  doi: 10.1016/j.fss.2009.06.005.  Google Scholar

[42]

L. Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems, 17 (1985), 313-328.  doi: 10.1016/0165-0114(85)90096-X.  Google Scholar

[43]

X. ZhangS. TanC.-C. Hang and P.-Z. Wang, An efficient computational algorithm for min-max operations, Fuzzy Sets and Systems, 104 (1999), 297-304.  doi: 10.1016/S0165-0114(97)00207-8.  Google Scholar

Figure 1.  Approximation to $ f $ given in (5) by pseudo-linear discrete operators
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