Abstract
In the present paper, we firstly give Jackson type estimates of approximation by max-product Shepard operators on \([-1,1]\), which combine pointwise and global estimates as a whole. Secondly, we investigate the weighted approximation by modified max-product Shepard operator for functions with singularities at the endpoints. Finally, we generalize max-product Shepard operators to the bounded convex set \(\mathbf \Omega \) on the scattered data, and obtain Jackson type estimates of approximation.
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References
Amato, U., Della Vecchia, B.: Rational operators based on q-integers. Results Math. 72, 1109–1128 (2017)
Amato, U., Della Vecchia, B.: New results on rational approximation. Results Math. 67, 345–364 (2015)
Bede, B., Coroianu, L., Gal, S.G.: Approximation by Max-Product Type Operators. Springer, Cham (2016)
Bede, B., Nobuhara, H., Hirota, K.: Max Product Shepard approximation operators. J. Adv. Comput. Intell. Intell. Inform. 10, 494–497 (2006)
Bede, B., Nobuhara, H., Dankova, M., Di Nola, A.: Approximation by pseudo-linear operators. Fuzzy Sets Syst. 159, 804–820 (2008)
Bede, B., Schwab, E.D., Nobuhara, H., Rudas, Imre J.: Approximation by Shepard type pseudo-linear operators and applications to image processing. Inter. J. Approx. Reason. 50, 21–36 (2009)
Coman, Gh., Trimbitas, R.: Combined Shepard univariate operators. East J. Approx. 7, 47–483 (2001)
Coroianu, L., Costarelli, D., Gal, S.G., Vinti, G.: The max-product generalized sampling operators: convergence and quantitative estimates. Appl. Math. Comput. 355, 173–183 (2019)
Costarelli, D., Sambucini, A.R.: Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators. Results Math. 73, 15 (2018)
Costarelli, D., Sambucini, A.R., Vinti, G.: Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications. Neural Comput. Appl. 31, 5069–5078 (2019)
Costarelli, D., Vinti, G.: Max-product neural network and quasi-interpolation operators activated by sigmoidal functions. J. Approx. Theory 209, 1–22 (2016)
Costarelli, D., Vinti, G.: Estimates for the neural network operators of the max-product type with continuous and p-integrable functions. Results Math. 73(1), 10–12 (2018)
Criscuolo, G., Mastroianni, G.: Estimate of the Shepard interpolatory procedure. Acta Math. Hungar. 61, 79–91 (1993)
Della Vechhia, B.: Direct and converse results by rational operators. Constr. Approx. 12, 271–285 (1996)
Della Vechhia, B.: Weighted approximation by rational operators. Result. Math. 43, 79–87 (2003)
Della Vechhia, B.: Direct and converse results for the weighted rational approximation of functions with inner singularities. J. Approx. Theory 126, 16–35 (2004)
Della Vechhia, B., Mastroianni, G.: Pointwise simultaneous approximation by rational operators. J. Approx. Theory 65, 140–150 (1991)
Della Vechhia, B., Mastroianni, G., Szabados, J.: Weighted uniform approximation on the semiaxis by rational operators. J. Inequal. Appl. 4, 241–264 (1999)
Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer Verlag, Berlin (1987)
Duman, O.: Statistical convergence of max-product approximating operators. Turk. J. Math. 34, 501–514 (2010)
Duman, O., Della Vechhia, B.: Complex Shepard operators and their summability. Results Math. 76 Paper No. 214, pp. 19 (2021)
Farwig, R.: Rate of convergence of Shepard’s global interpolation formula. Math. Comp. 46, 577–590 (1986)
Mastroianni, G., Szabados, J.: Balazs-Shepard operators on infinite intervals, II. J. Approx. Theory 90, 1–8 (1997)
Szabados, J.: Direct and converse approximation theorems for the Shepard operators. Approx. Theory Appl. 7, 63–76 (1991)
Z. M. Wu, Scattered Data Approximation: Theories, Models, and Algorithms, China Science Publishing, (2007). (in Chinese)
Xiao, W., Zhou, S.P.: A Jackson type estimate for Shepard operators in \(L^p\) spaces for \(p\ge 1\). Acta Math Hungar. 95, 217–224 (2002)
Xie, T.F., Zhang, R.J., Zhou, S.P.: Three conjectures on Shepard operators. J. Approx. Theory 93, 399–414 (1998)
Yu, D.S.: On weighted approximation by rational operators for functions with singularities. Acta Math. Hungar. 136, 56–75 (2012)
Yu, D.S.: On approximation by Shepard-Lagrange operators. Acta Math. Sninca 53, 97–108 (2010)
Yu, D.S., Zhou, S.P.: Approximation by rational operators in \(L^p\) spaces. Math. Nachr. 282, 1600–1618 (2009)
Zhou, X.L.: The saturation class of Shepard operators. Acta Math Hungar. 80, 293–310 (1998)
Funding
The work is partially supported by Zhejiang Provincial Natural Science Foundation (LR19F020005), The National Key Research and Development Program of China (2018YFB0104503), National Natural Science Foundation of China (NSFC61602404, NSFC12271133) and Fundamental Research Funds for the Central Universities (2018FZA5014).
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Yu, D. On Approximation by Max-product Shepard Operators. Results Math 77, 219 (2022). https://doi.org/10.1007/s00025-022-01746-w
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DOI: https://doi.org/10.1007/s00025-022-01746-w