Approximation results for a general class of Kantorovich type operators

Advanced Nonlinear Studies

Volumn 14, Issue 4, 2014, Pages 991-1011

  Vinti, Gianluca ^a   Zampogni, Luca ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Estimates; Kantorovich convolution type operators; Kantorovich Mellin type operators; Kantorovich sampling type operators; Modular convergence; Orlicz spaces; Pointwise convergence; Uniform convergence

Indexed keywords

EID: 84910058049     PISSN: 15361365     EISSN: None     Source Type: Journal    
DOI: 10.1515/ans-2014-0410     Document Type: Article

References (39)

1. L. Angeloni and G. Vinti, A unified approach to approximation results with applications to nonlinear sampling theory, Int. J. Math. Sci. 3, no. 1, (2004), 93-128.
2. L. Angeloni and G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing, Differential and Integral Equations 18, no. 8, (2005), 855-890.
3. C. Bardaro, P. L. Butzer, R.L. Stens and G. Vinti, Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis (Munich) 23, no. 4, (2003), 299-340.
4. C. Bardaro, P. L. Butzer, R.L. Stens and G. Vinti, Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl. 316, no. 1, (2006), 269-306.
5. C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampling Theory in Signal and Image Processing 6, no. 1, (2007), 29-52.
6. C. Bardaro, P. L. Butzer, R.L. Stens and G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Transaction on Information Theory, 56 (1) (2010), 614-633.
7. C. Bardaro and I. Mantellini, A modular convergence theorem for general nonlinear integral operators, Comment. Math. 36 (1996), 27-37.
8. C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich type discrete operators, Numerical Functional Analysis and Optimization 33 (4), (2012), 374-396.
9. C. Bardaro and I. Mantellini, Pointwise convergence theorems for nonlinear Mellin Convolution Operators, Int. J. Pure Appl. Math. 27 (4) (2006), 431-447.
10. C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, de Gruyter Series in Nonlinear Analysis and Applications, vol. 9, Walter de Gruyter & Co., Berlin, 2003.
11. C. Bardaro and G. Vinti, A general approach to the convergence theorems of generalized sampling series, Appl. Anal. 64 (1997), 203-217.
12. C. Bardaro and G. Vinti, Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator, Results Math. 34 (1998), 224-240, special issue dedicated to Prof. P.L. Butzer.
13. C. Bardaro and G. Vinti, An Abstract Approach to Sampling Type Operators Inspired by the Work of P.L. Butzer - Part I - Linear Operators, Sampling Theory in Signal and Image Processing, 2 3 (2003), 271-296.
14. P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), 185-212.
15. P.L. Butzer, A. Fisher and R.L. Stens, Generalized sampling approximation of multivariate signals: general theory, Atti Sem. Mat. Fis. Univ. Modena, 41,(1) (1993), 17-37.
16. P.L. Butzer and S. Jansche, A direct approach to the Mellin Transform, J. Fourier Anal. Appl., 3 (1997), 325-376.
17. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag, Basel, and Academic Press, New York, 1971.
18. P. L. Butzer, S. Ries and R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory 50 (1987), 25-39.
19. P. L. Butzer and R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev. 34 (1992), 40-53.
20. P. L. Butzer and R.L. Stens, Linear Prediction by Samples from the Past, In "Advanced Topics in Shannon Sampling and Interpolation Theory", R. J. Marks II ed., Springer Texts Electrical Engrg., 157-183, Springer, New York 1993.
21. D. Costarelli and G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special volume dedicated to Prof. Giovanni Prodi, (9) IV (2011), 445-468.
22. D. Costarelli and G. Vinti, Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing, Numerical Functional Analysis and Optimization 34 (8) (2013), 819-844.
23. D. E. Edmunds and M.Krbec, Two limiting cases of Sobolev imbeddings, Houston J. Math. 21, no. 1, (1995), 119-128.
24. L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). viii+268 pp. ISBN: 0-8493-7157-0
25. S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal. 204, no. 1, (2003), 196-227.
26. L.V. Kantorovich, Sur certains développements suivant les polynomes de la forme de S. Bernstein. I, C.R. Acad. Sc. URSS (1930), 563-568 (Russian).
27. M. A. Krasnosel'skiѤ and Ja. B. RutickiѤ, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.
28. H. Mascart, Misura di Haar su gruppi topologici, Editrice Universitaria Levrotto e Bella, 1983, Torino.
29. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
30. rd ed. New York, Gordon and Breach Science Publishers, 1986.
31. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991.
32. M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002.
33. L. L. Schumaker, Spline Functions: Basic Theory, New York: Wiley 1981.
34. E. M. Stein, Note on the class L log L, Studia Math. 32 (1969), 305-310.
35. R. L. Stens, Sampling with generalized kernels, In: Sampling Theory and Signal Analysis: Advanced Topics (Higgins, J.R. and Stens, R.L., eds.), Oxford Science Publications, Oxford University Press, Oxford (1999), 130-157.
36. G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Applicable Analysis, 79 (2001), 217-238.
37. G. Vinti, Approximation in Orlicz spaces for linear integral operators and applications, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 76 (2005), 103-127.
38. G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, Journal of Approximation Theory 161 (2009), 511-528.
39. G. Vinti - L. Zampogni, A Unifying Approach to Convergence of Linear Sampling Type Operators in Orlicz Spaces, Advances in Differential Equations 16, numbers 5-6, (2011), 573-600.