Approximation by sampling type discrete operators

Year 2020, Volume: 69 Issue: 1, 969 - 980, 30.06.2020

İsmail Aslan

https://doi.org/10.31801/cfsuasmas.671237

Cited By: 5

Abstract

In this paper, we deal with discrete operators of sampling type. It is known that this type of operators are related to generalized sampling series and they have important applications. In this work, using bounded and uniformly continuous functions we get general estimations under usual supremum norm with the help of summability method. We also study the degree of approximation with respect to suitable Lipschitz class of continuous functions. Finally, we give specific kernels which verify our kernel assumptions.

Keywords

Sampling type operators, rate of approximation, summability process

References

• Aslan, İ., Duman, O., A summability process on Baskakov-type approximation, Period. Math. Hungar., 72 (2) (2016), 186--199.
• Aslan, I., Duman, O., Summability on Mellin-type nonlinear integral operators, Integral Transform, Spec. Funct., 30 (6) (2019), 492-511.
• Aslan, I., Duman, O., Approximation by nonlinear integral operators via summability process, Math. Nachr., 293 (3) (2020), 430-448.
• Aslan, I., Duman, O., Characterization of absolute and uniform continuity, Hacet. j. math. stat., (2020), (in press) doi: 10.15672/hujms.585581.
• Aslan, I., Duman, O., Nonlinear approximation in N-dimension with the help of summability methods (submitted).
• Atlihan, Ö.G., Orhan, C., Summation process of positive linear operators, Comput Math Appl., 56 (2008), 1188--1195.
• Bell, H. T., A-summability, Dissertation, Lehigh University, Bethlehem, Pa., 1971.
• Bell, H. T., Order summability and almost convergence, Proc. Amer. Math. Soc., 38 (1973), 548--552.
• Bezuglaya, L., Katsnelson, V., The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend., 12, (1994), 511-534.
• 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.
• 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.
• 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.
• Butzer, P.L., Stens, R.L., Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev., 34 (1) (1992), 40-53.
• Butzer, P.L., Splettstösser, W., Stens, R.L., The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein. 90 (1988), 1-70.
• Gokcer, T.Y., Duman, O., Summation process by max-product operators, Computational Analysis, AMAT 2015, Univ. Econ. & Technol. Ankara, Turkey, (2016), 59-67.
• Hardy, G. H., Divergent series, Oxford Univ. Press, London, 1949.
• Jurkat, W. B., Peyerimhoff, A., Fourier effectiveness and order summability, J. Approx. Theory, 4 (1971), 231--244.
• Jurkat, W. B., Peyerimhoff, A., Inclusion theorems and order summability, J. Approx. Theory, 4 (1971), 245--262.
• Keagy, T. A., Ford, W. F., Acceleration by subsequence transformations, Pacific J. Math., 132 (2) (1988), 357--362.
• Küçük, N., Duman, O., Summability methods in weighted approximation to derivatives of functions, Serdica Math. J., 41 (4) (2015), 355-368.
• Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., 80, (1948), 167--190.
• Mohapatra, R. N., Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory, no. 20 (1977), 239--250
• 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), 746-756.
• Royden, H.L., Fitzpatrick, P.M., Real Analysis (4th edition), Pearson Education, 2010.
• Smith, D. A., Ford, W. F., Acceleration of linear and logarithmical convergence, Siam J. Numer. Anal., 16 (1979), 223--240.
• Swetits, J. J., On summability and positive linear operators, J. Approx. Theory, 25 (1979), 186--188.
• Wimp, J., Sequence Transformations and Their Applications, Academic Press, New York, 1981.