A unifying approach to convergence of linear sampling type operators in Orlicz spaces

Advances in Differential Equations

Volumn 16, Issue 5-6, 2011, Pages 573-600

  Vinti, Gianluca ^a   Zampogni, Luca ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

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EID: 84874638760     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (53)

1. L. Angeloni and G. Vinti, A unified approach to approximation results with applications to nonlinear sampling theory, Int. J. Math. Sci., 3 (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 (2005), 855- 890.
3. 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 (2006), 269-306.
4. 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.
5. C. Bardaro and I. Mantellini, A modular convergence theorem for general nonlinear integral operators, Comment. Math. Prace Mat. 36 (1996), 27-37.
6. C. Bardaro and I. Mantellini, On global approximation properties of discrete operators in Orlicz spaces, JIPAM. J. Inequal. Pure Appl. Math. 6 No. 4 (2005), Article 123, 34 pp. (electronic).
7. 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.
8. C. Bardaro and G. Vinti, A general approach to the convergence theorems of generalized sampling series, Appl. Anal. 64 (1997), 203-217.
9. 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.
10. 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 No.3 (2003), 271-295.
11. C. Bardaro and G. Vinti, An abstract approach to sampling type operators inspired by the work of P.L. Butzer. Part II Nonlinear operators, Sampling Theory in Signal and Image Processing 3 No. 1 (2004), 29-44.
12. M. G. Beaty and M. M. Dodson, Abstract harmonic analysis and the sampling theorem, in "Sampling Theory in Fourier and Signal Analysis: Advanced Topics," Oxford Science Publications, J. R. Higgings and R. L. Stens eds., Oxford Univ. Press, Oxford 1999.
13. C. Bennett and K. Rudnick, On Lorentz-Zygmund Spaces, Dissertationes Math. (Rozprawy Mat.) 175 (1980).
14. L. Beluglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwendungen 12 (1993), 511-534.
15. P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), 185-212.
16. P. L. Butzer and G. Hinsen, Reconstruction of bounded signal from pseudoperiodic, irregularly spaced samples, Signal Process. 17 (1989), 1-17.
17. P. L. Butzer and J. Lei, Errors in truncated sampling series with measured sampled values for not-necessarily bandlimited functions, Funct. Approx. Comment. Math. 26 (1998), 25-39.
18. P. L. Butzer and J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 No.2 (2000), 245-255.
19. P. L. Butzer, J. Mawhin and P. Vetro, eds., "Charles-Jean de La Vallée Poussin, Collected works/Oeuvres scientifiques. Vol. III: Approximation Theory, Fourier Analysis, Quasi-Analytic Functions," Académie Royale de Belgique, Brussels, and Circolo Matematico di Palermo, Palermo, 2004.
20. P. L. Butzer and R. J. Nessel, "Fourier Analysis and Approximation," Birkhäuser Verlag, Basel, and Academic Press, New York, 1971.
21. 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.
22. P. L. Butzer, G. Schmeisser and R. L. Stens, An introduction to sampling analysis, In: Nonuniform Sampling, Theory and Practice (Marvasti, F., ed.), Information Technology: Transmission, Processing and Storage, Kluwer Academic/Plenum Publishers, New York (2001), 17-121.
23. 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.
24. P. L. Butzer and R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev. 34 (1992), 40-53.
25. 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.
26. M. M. Dodson and A. M. Silva, Fourier analysis and the sampling theorem, Proc. Royal Irish Acad. Sect. A 85 (1985), 81-108.
27. D. E. Edmunds and M. Krbec, Two limiting cases of Sobolev imbeddings, Houston J. Math. 21 No.1 (1995), 119-128.
28. A. Fiorenza, Some remarks on Stein's L log L result, Differential Integral Equations 5 No. 6 (1992), 1355-1362.
29. K. Gröchenig, Reconstruction algorithms in irregular sampling, Math. Comp. 59 No. 199 (1992), 181-194.
30. S. Haber and O. Shisha, Improper integrals, simple integrals and numerical quadrature, J. Approx. Theory 11 (1974), 1-15.
31. S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal. 204 No. 1 (2003), 196-227.
32. J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12 (1985), 45-89.
33. J. R. Higgins, "Sampling Theory in Fourier and Signal Analysis: Foundations," Oxford Univ. Press, Oxford 1996.
34. J. R. Higgins and R. L. Stens Eds, "Sampling Theory in Fourier and Signal Analysis: Advanced Topics," Oxford Univ. Press, Oxford 1999.
35. A. J. Jerri, The Shannon sampling - its various extensions and applications: a tutorial review, Proc: IEEE 65 (1977), 1565-1596.
36. L. V. Kantorovich, Sur certains développements suivant les polynomes de la forme de S. Bernstein I,II, C. R. Acad. Sc. URSS (1930), 563-568 (Russian).
37. V. A. Kotel'nikov, On the carrying capacity of "ether" and wire in elettrocommunications, In "Material for the First All-Union Conference on Questions of Communications", Izd. Red. Upr. Svyazi RKKA, Moscow, 1933 (in Russian); English translation in "Appl. Numer. Harmon. Anal. Modern Sampling Theory", pp. 27-45, Birkhäuser, Boston, MA, 2001.
38. M. A. Krasnosel'skiǐ and Ja. B. Rutickiǐ, "Convex Functions and Orlicz Spaces," P. Noordhoff Ltd., Groningen, 1961.
39. I. Mantellini and G. Vinti, Approximation results for nonlinear integral operators in modular spaces and applications, Annales Polonici Mathematici, 81 No. 1 (2003), 55-71.
40. J. Musielak, "Orlicz Spaces and Modular Spaces," Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
41. 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.
42. 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.
43. S. Ries and R. L. Stens, Approximation by generalized sampling series, In: Constructive Theory of Functions, Proc. Conf., Varna, Bulgaria, 1984 (Sendov, Bl., Petrushev, P., Maleev, R., and Tashev, S., eds.), Publishing House of the Bulgarian Academy of Sciences, Sofia, 1984, 746-756.
44. H. J. Schmeisser and W. Sickel, Sampling theory and function spaces, In: Applied Mathematics Reviews (Anastassiou, G. A., ed.), vol. 1, World Scientific Publishing Co. Inc., River Edge, NJ (2000), 205-284.
45. C. E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10-21.
46. E. M. Stein, Note on the class L log L, Studia Math. 32 (1969), 305-310.
47. 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.
48. H. Triebel, "The Structure of Functions," Birkhäuser Verlag, Basel, 2001.
49. C. Vinti, A Survey on Recent Results of the Mathematical Seminar in Perugia, inspired by the Work of Professor P. L. Butzer, Results Math. 34 (1998), 32-55.
50. G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Appl. Anal. 79 No. 1 & 2 (2001), 217-238.
51. G. Vinti, Approximation in Orlicz spaces for linear integral operators and applications, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 76 (2005), 103-127.
52. G. Vinti, L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, Jour. of Approx. Theory, 161 (2009), 511-528.
53. E. T. Whittaker, On the functions which are represented by the expansion of the interpolation theory, Proc. Royal Soc. Edinburgh 35 (1915), 181-194.