Modular filter convergence theorems for abstract sampling type operators

Applicable Analysis

Volumn 92, Issue 11, 2013, Pages 2404-2423

  Bardaro, C ^a   Boccuto, A ^a   Dimitriou, X ^b   Mantellini, I ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

^b UNIVERSITY OF ATHENS   (Greece)

Author keywords

filter convergence; filter convergence in measure; filter exhaustiveness; filter singularity; integral operator; modular; modular convergence; sampling series

Indexed keywords

EID: 84887967314     PISSN: 00036811     EISSN: 1563504X     Source Type: Journal    
DOI: 10.1080/00036811.2012.738480     Document Type: Article

References (34)

1. Boccuto, A and Dimitriou, X. 2012. Modular filter convergence theorems in Orlicz spaces for Urysohn integral operators and applications. Acta Math. Sin. (Engl. Ser.), (in print)
2. Boccuto, A and Dimitriou, X. 2012. Modular convergence theorems for integral operators in the context of filter exhaustiveness and applications. Med. J. Math., published on-line DOI 10.1007/s00009-012-0199-z
3. Bardaro, C and Mantellini, I. 2005. On global approximation properties of abstract integral operators in Orlicz spaces and applications. J. Inequal. Pure Appl. Math., 6 (4) Article 123
4. Bardaro, C and Mantellini, I. 2006. Approximation properties in abstract modular spaces for a class of general sampling type operators. Appl. Anal., 85 (4): 383-413.
5. Bardaro, C, Musielak, J and Vinti, G. 2003. Nonlinear Integral Operators and Applications, Berlin: de Gruyter.
6. Kozlowski, WM. 1988. Modular Function Spaces, Monographs and Text Books in Pure and Applied Mathematics, New York: Marcel Dekker.
7. Musielak, J. 1983. Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, New York: Springer-Verlag.
8. Fast, H. 1951. Sur la convergence statistique. Colloq. Math., 2: 241-244.
9. Steinhaus, H. 1951. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 2: 73-74.
10. Agratini, O. 2011. Statistical convergence of a non-positive approximation process. Chaos Solitons Fractals, 44 (11): 977-981.
11. Anastassiou, GA and Duman, O. 2011. Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent Systems Reference Library, Vol. 14, Berlin: Springer-Verlag.
12. Butzer, PL, Fischer, A and Stens, RL. 1993. Generalized sampling approximation of multivariate signals: General theory. Atti Sem. Mat. Fis. Univ. Modena, 41: 17-37.
13. Butzer, PL and Stens, RL. 1992. Sampling theory for not necessarily band-limited functions: An historical overview. SIAM Rev., 34: 40-53.
14. p(ℝ)-spaces by generalized sampling series based on linear combinations of B-splines. Integral Transforms Spec. Funct., 19 (1-2): 35-58.
15. Fischer, A and Stens, RL. 1990. Generalized sampling approximation of multivariate signals: Inverse approximation theorems. Colloq. Math. Soc. Janos Bolyai, 58: 275-286. (Approximation Theory, Kecskemet, Hungary)
16. p-metrics. Sampl. Theory Signal Image Process., 5 (1): 59-87.
17. Feuer, A and Goodwin, GC. 2005. Reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples. IEEE Trans. Signal Process., 53 (11): 4273-4282.
18. Bardaro, C, Butzer, PL, Stens, RL and Vinti, G. 2010. Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inform. Theory, 56 (1): 614-633.
19. Kuratowski, K. 1966/1968. Topology, Vols. I and II, New York, London: Academic Press.
20. Demirci, K. 2001. ℐ-limit superior and limit inferior. Math. Commun., 6: 165-172.
21. Bardaro, C and Mantellini, I. 2006. Uniform modular integrability and convergence properties for a class of Urysohn integral operators in function spaces. Math. Slovaca, 56 (4): 465-482.
22. Rao, MM and Ren, ZD. 1991. Theory of Orlicz Spaces, New York: Marcel Dekker.
23. Mantellini, I. 1998. Generalized sampling operators in modular spaces. Comment. Math., 38: 77-92.
24. Kostyrko, P, Šalát, T and Wilczyński, W. 2000/2001. I-convergence. Real Anal. Exchange, 26: 669-685.
25. Haber, S and Shisha, O. 1974. Improper integrals, simple integrals and numerical quadrature. J. Approx. Theory, 11: 1-15.
26. Bardaro, C, Butzer, PL, Stens, RL and Vinti, G. 2007. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process., 6 (1): 29-52.
27. S. Ries and R.L. Stens, Approximation by generalized sampling series, in Constructive Theory of Functions, B.I. Sendov, P. Petrushev, R. Maleev, and S. Tashev, eds., Proc. Conf. Varna, Bulgaria, 1984, Publishing House of the Bulgarian Academy of Sciences, Sofia, 1984, pp. 746-756
28. Angeloni, L and Vinti, G. 2005. Rate of approximation for nonlinear integral operators with application to signal processing. Differ. Integral Eqns, 18 (8): 855-890.
29. Holmes, RB. 1979. Mathematical foundations of signal processing. SIAM Rev., 21 (3): 361-388.
30. Vinti, G. 2001. A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal., 79: 217-238.
31. Bardaro, C and Mantellini, I. 2010. Generalized sampling approximation of bivariate signals: rate of pointwise convergence. Num. Funct. Anal. Optim., 31 (2): 131-154.
32. Bardaro, C and Mantellini, I. 2011. Bivariate Mellin convolution operators: Quantitative approximation theorems. Math. Comput. Model., 53: 1197-1207.
33. Bardaro, C and Mantellini, I. 2011. Asymptotic behaviour of Mellin-Féjer convolution operators. East J. Approx., 17 (2): 181-201.
34. Mantellini, I and Vinti, G. 1996. Modular estimates for nonlinear integral operators and applications in fractional calculus. Numer. Funct. Anal. Optimiz., 17 (1-2): 143-165.