Nonlinear approximation by $N$-dimensional sampling type discrete operators with applications

Abstract

In this paper, we explore $N$-dimensional nonlinear discrete operators, closely related to generalized sampling series. We investigate their approximation properties by using the supremum norm and employ a summability method to generalize the discrete operators. The order of convergence is studied by using suitable Lipschitz classes of uniformly continuous functions. We exemplify kernel functions that meet the necessary conditions. Additionally, in the final section of the paper, we propose an operator-based method for digital image zooming.

Keywords

• Nonlinear operators
• discrete operators
• sampling type operators
• order of convergence
• summability process
• digital image processing

References

1. Acu, A. M., Hodis, L., Raşa, I., Multivariate weighted Kantorovich operators, Math. Found. Comput., 3(2) (2020), 117-124. https://doi.org/10.3934/mfc.2020009
2. Angeloni, L., Vinti, G., Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math., 49 (2006), 1-23. https://doi.org/10.1007/s00025-006-0208-2
3. Angeloni, L., Vinti, G., Erratum to: Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math., 57 (2010), 387-391. https://doi.org/10.1007/s00025-010-0019-3
4. Angeloni, L., Vinti, G., Discrete operators of sampling type and approximation in φ-variation, Math. Nachr., 291(4) (2018), 546-555. https://doi.org/10.1002/mana.201600508
5. Angeloni, L., Costarelli, D., Vinti, G., A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767. https://doi.org/10.5186/aasfm.2018.4343
6. Aslan, İ., Approximation by sampling type discrete operators, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(1) (2020), 969-980. https://doi.org/10.31801/cfsuasmas.671237
7. Aslan, İ., Approximation by sampling-type nonlinear discrete operators in phi-variation, Filomat, 35(8) (2021), 2731-2746. https://doi.org/10.2298/fil2108731a
8. Aslan, İ., Duman, O., A summability process on Baskakov-type approximation, Period. Math. Hungar., 72(2) (2016), 186-199. https://doi.org/10.1007/s10998-016-0120-9

9. Aslan, İ., Duman, O., Approximation by nonlinear integral operators via summability process, Math. Nachr., 293(3) (2020), 430-448. https://doi.org/10.1002/mana.201800187
10. Aslan, İ., Duman, O., Summability on Mellin-type nonlinear integral operators, Integral Transform. Spec. Funct., 30(6) (2019), 492-511. https://doi.org/10.1080/10652469.2019.1594209
11. Aslan, İ., Duman, O., Characterization of absolute and uniform continuity, Hacet. J. Math. Stat., 49(5) (2020), 1550-1565. https://doi.org/10.15672/hujms.585581
12. Bardaro, C., Butzer, P. L., Stens, R. L., Vinti, G., Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process., 6 (2007), 29-52. https://doi.org/10.1007/bf03549462
13. Bardaro, C., Butzer, P. L., Stens, R. L., Vinti, G., Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Inform. Theory, 56(1) (2010), 614-633. https://doi.org/10.1109/TIT.2009.2034793
14. Bardaro, C., Vinti, G., A general approach to the convergence theorems of generalized sampling series, Appl. Anal., 64 (1997), 203-217. https://doi.org/10.1080/00036819708840531
15. Bede, B., Schwab, E. D., Nobuhara, H., Rudas, I. J., Approximation by Shepard type pseudolinear operators and applications to image processing, Int. J. Approx. Reason., 1(50) (2009), 21-36. https://doi.org/10.1016/j.ijar.2008.01.007
16. Bell, H. T., A-Summability, Dissertation, (Lehigh University, Bethlehem, Pa., 1971).
17. Bell, H. T., Order summability and almost convergence, Proc. Amer. Math. Soc., 38 (1973), 548-552.
18. Bezuglaya L., Katsnelson, V., The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend., 12 (1994), 511-534.
19. Boccuto, A., Dimitriou, X., Rates of approximation for general sampling-type operators in the setting of filter convergence, Appl. Math. Comput., 229 (2014), 214-226. https://doi.org/10.1016/j.amc.2013.12.044
20. Butzer P. L., Stens, R. L., Prediction of non-bandlimited signals from past samples in terms of splines of low degree, Math. Nachr., 132 (1987), 115-130. https://doi.org/10.1002/mana.19871320109
21. Butzer P. L., Stens, R. L., Linear predictions in terms of samples from the past: an overview, Proceedings of Conference on Numerical Methods and Approximation Theory III (G. V. Milovanovic, ed.), University of Nis, Yugoslavia, (1988), 1-22.
22. Butzer, P. L., Splettst¨osser, W., Stens, R. L., The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein, 90 (1988), 1-70.
23. Butzer, P. L., Stens, R. L., Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev., 34(1) (1992), 40-53. https://doi.org/10.1137/1034002
24. Costarelli, D., Piconi, M., Vinti, G., The multivariate Durrmeyer-sampling type operators in functional spaces. Dolomites Res. Notes Approx., 15(5) (2022), 128-144. https://doi.org/10.14658/PUPJ-DRNA-2022-5-11
25. Costarelli, D., Vinti, G., Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34(8) (2013), 819-844. https://doi.org/10.1080/01630563.2013.767833
26. Gökçer, T. Y., Aslan, İ., Approximation by Kantorovich-type max-min operators and its applications, Appl. Math. Comput., 423, (2022), 127011. https://doi.org/10.1016/j.amc.2022.127011
27. Gökçer, T. Y., Duman, O., Summation process by max-product operators, Computational Analysis, AMAT 2015, Univ. Econ. & Technol. Ankara, Turkey, (2016), pp. 59-67. https://doi.org/10.1007/978-3-319-28443-9 4
28. Hardy, G. H., Divergent Series, Oxford Univ. Press, London, 1949.
29. Jurkat, W. B., Peyerimhoff, A., Fourier effectiveness and order summability, J. Approx. Theory, 4 (1971), 231-244. https://doi.org/10.1016/0021-9045(71)90011-6
30. Jurkat, W. B., Peyerimhoff, A., Inclusion theorems and order summability, J. Approx. Theory, 4 (1971), 245-262. https://doi.org/10.1016/0021-9045(71)90012-8
31. Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., 80 (1948), 167-190. https://doi.org/10.1007/BF02393648
32. Mantellini, I., Vinti, G., Approximation results for nonlinear integral operators in modular spaces and applications, Ann. Polon. Math., 81(1) (2003), 55-71. https://doi.org/10.4064/ap81-1-5
33. Mohapatra, R. N., Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory, 20 (1977), 239-250. https://doi.org/10.1016/0021-9045(77)90058-2
34. Ries, S., Stens, R. L., Approximation by generalized sampling series, Proceedings of the International Conference on Constructive Theory of Functions (Varna, 1984), Bulgarian Academy of Science, Sofia, 1984, pp. 746-756.
35. Smith, D. A., Ford, W. F., Acceleration of linear and logarithmical convergence, Siam J. Numer. Anal., 16 (1979), 223-240. https://doi.org/10.1137/0716017
36. Stieglitz, M., Eine verallgemeinerung des begriffs der fastkonvergenz, Math. Japon., 18(1) (1973), 53-70.
37. Swetits, J. J., Note: On summability and positive linear operators, J. Approx. Theory, 25(2) (1979), 186-188. https://doi.org/10.1016/0021-9045(79)90008-x
38. Turan, C., Duman, O., Statistical convergence on timescales and its characterizations, In Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012, (2013), 57-71, Springer New York. https://doi.org/10.1007/978-1-4614-6393-1 3
39. Wimp, J., Sequence Transformations and Their Applications, Academic Press, New York, 1981.