# American Institute of Mathematical Sciences

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. Acar, D. 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. Angeloni, D. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] B. Bede, H. Nobuhara, M. 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. [14] B. Bede, E. D. Schwab, H. 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. [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. [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. [17] P. L. Butzer, W. Splettstösser and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70. [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. [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. [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. [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. [22] L. Coroianu, D. Costarelli, S. 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. [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. [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. [25] L. Coroianu, S. 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. [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. [27] D. Costarelli, A. 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. [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. [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. [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. [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. [32] O. Duman, Statistical convergence of max-product approximating operators, Turkish J. Math., 34 (2010), 501-514. [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. [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. [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. [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. [37] C.-C. Liu, Y.-K. Wu, Y.-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. [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. [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. [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. [41] H. Tahayori, A. G. B. Tettamanzi, G. 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. [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. [43] X. Zhang, S. Tan, C.-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.

show all references

##### References:
 [1] T. Acar, D. 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. Angeloni, D. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] B. Bede, H. Nobuhara, M. 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. [14] B. Bede, E. D. Schwab, H. 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. [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. [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. [17] P. L. Butzer, W. Splettstösser and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70. [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. [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. [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. [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. [22] L. Coroianu, D. Costarelli, S. 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. [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. [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. [25] L. Coroianu, S. 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. [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. [27] D. Costarelli, A. 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. [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. [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. [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. [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. [32] O. Duman, Statistical convergence of max-product approximating operators, Turkish J. Math., 34 (2010), 501-514. [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. [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. [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. [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. [37] C.-C. Liu, Y.-K. Wu, Y.-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. [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. [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. [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. [41] H. Tahayori, A. G. B. Tettamanzi, G. 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. [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. [43] X. Zhang, S. Tan, C.-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.
Approximation to $f$ given in (5) by pseudo-linear discrete operators
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