Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes

Journal of Mathematical Analysis and Applications

Volumn 419, Issue 2, 2014, Pages 818-838

  Boccuto, Antonio ^a   Candeloro, Domenico ^a   Sambucini, Anna Rita ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Filter convergence; It integral; Modular; Moment operator; Vector lattice; Vitali convergence theorem

Indexed keywords

EID: 84903472025     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2014.05.014     Document Type: Article

References (44)

1. Angeloni L., Vinti G. Rate of approximation for nonlinear integral operators with application to signal processing. Differential Integral Equations 2005, 18(8):855-890.
2. φ-spaces in multidimensional setting. J. Math. Anal. Appl. 2009, 349(2):317-334.
3. Angeloni L., Vinti G. Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators. Numer. Funct. Anal. Optim. 2010, 31:519-548. 10.1080/01630563.2010.490549.
4. Angeloni L., Vinti G. Approximation in variation by homothetic operators in multidimensional setting. Differential Integral Equations 2013, 26(5/6):655-674.
5. Angeloni L., Vinti G. Convergence and rate of approximation in BVφ(R+n) for a class of Mellin integral operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 2014, 25:1-16. 10.4171/RLM.
6. Bardaro C., Boccuto A., Dimitriou X., Mantellini I. Modular filter convergence theorems for abstract sampling-type operators. Appl. Anal. 2013, 92(11):2404-2432.
7. 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. 2006, 6:29-52.
8. Bardaro C., Butzer P.L., Stens R.L., Vinti G. Approximation of the Whittaker sampling series in terms of an average modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 2007, 316:269-306.
9. 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 2010, 56(1):614-633.
10. Bardaro C., Mantellini I. Uniform modular integrability and convergence properties for a class of Urysohn integral operators in function spaces. Math. Slovaca 2006, 56(4):465-482.
11. Bardaro C., Mantellini I. Approximation properties in abstract modular spaces for a class of general sampling type operators. Appl. Anal. 2006, 85(4):383-413.
12. Bardaro C., Mantellini I. On Mellin convolution operators: a direct approach to the asymptotic formulae. Integral Transforms Spec. Funct. 2014, 25(3):182-195.
13. Bardaro C., Musielak J., Vinti G. Nonlinear Integral Operators and Applications. de Gruyter Ser. Nonlinear Anal. Appl. 2003, vol. 9. Walter de Gruyter & Co., Berlin.
14. Bardaro C., Sciamannini S., Vinti G. Convergence in BVφ by nonlinear Mellin-type convolution operators. Funct. Approx. Comment. Math. 2001, 29:17-28.
15. Bardaro C., Vinti G. Integral operators on vector measures and applications to the moment kernel. Rend. Mat. Appl. (7) 1988, 8:149-164.
16. Bardaro C., Vinti G. Modular convergence in generalized Orlicz spaces for moment type operators. Appl. Anal. 1989, 32:265-276.
17. Bardaro C., Vinti G. On convergence of moment operators with respect to the φ-variation. Appl. Anal. 1991, 41:247-256.
18. Bardaro C., Vinti G. A general approach to the convergence theorems of generalized sampling series. Appl. Anal. 1997, 64(3-4):203-217.
19. Boccuto A., Candeloro D. Integral and ideals in Riesz spaces. Inform. Sci. 2009, 179:647-660.
20. Boccuto A., Candeloro D. Differential calculus in vector lattices and applications to g-calculus. Mediterr. J. Math. 2011, 8(3):315-329.
21. Boccuto A., Dimitriou X. Modular filter convergence theorems in Orlicz spaces for Urysohn integral operators and applications. Acta Math. Sin. (Engl. Ser.) 2013, 29(6):1055-1066. 10.1007/s10114-013-1443-6.
22. Boccuto A., Dimitriou X. Modular convergence theorems for integral operators in the context of filter exhaustiveness and applications. Mediterr. J. Math. 2013, 10(2):823-842. 10.1007/s00009-012-0199-z.
23. Boccuto A., Dimitriou X. Convergence Theorems for Lattice Group-Valued Measures 2014, Bentham Science Publ.
24. Boccuto A., Dimitriou X., Papanastassiou N. Schur lemma and limit theorems in lattice groups with respect to filters. Math. Slovaca 2012, 62(6):1145-1166.
25. Boccuto A., Minotti A.M., Sambucini A.R. Set-valued Kurzweil-Henstock integral in Riesz space setting. Panamer. Math. J. 2013, 23(1):57-74.
26. Boccuto A., Sambucini A.R. Abstract integration in convergence groups. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 1998, 46:315-333.
27. Breiman P. Probability 1968, Addison-Wesley Publishing Company, Reading, MA, London, Don Mills, ON.
28. Butzer P.L., Nessel R.J. Fourier Analysis and Approximation. Pure Appl. Math. 1971, vol. 40. Academic Press, New York, London.
29. Candeloro D., Sambucini A.R. Filter convergence and decompositions for vector lattice-valued measures. Mediterr. J. Math. 2014, preprint 2013, in press.
30. Choksi J.R. Vitali's convergence theorem on term by term integration. Enseign. Math. 2001, 47:269-285.
31. Costarelli D., Vinti G. Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. (8) 2011, 9(4):445-468.
32. Costarelli D., Vinti G. Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numer. Funct. Anal. Optim. 2013, 34(8):819-844.
33. Costarelli D., Vinti G. Order of approximation for sampling Kantorovich operators. J. Integral Equations Appl. 2014, in press.
34. Garsia A.M., Rodemich E., Rumsey H. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 1970/1971, 20:565-578.
35. Holmes R.B. Mathematical foundations of signal processing. SIAM Rev. 1979, 21(3):361-388.
36. Kozlowski W.M. Modular Function Spaces. Pure Appl. Math. 1988, Marcel Dekker, New York.
37. Musielak J. Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1983, vol. 1034. Springer-Verlag, New York.
38. Nualart D. Fractional Brownian motion: stochastic calculus and applications. Proc. International Congress of Mathematicians, vol. 3 2006, 1541-1562. EMS, Nr. 74.
39. Rao M.M., Ren Z.D. Theory of Orlicz Spaces 1991, Marcel Dekker, New York.
40. Varadhan S.R.S. Diffusion processes. Handbook of Statistics, vol. 19 2001, 853-872. Shandbhag, Rao (Eds.).
41. Vinti G. A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 2001, 79:217-238.
42. Vinti G., Zampogni L. Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces. J. Approx. Theory 2009, 161(2):511-528.
43. Vinti G., Zampogni L. A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Adv. Differential Equations 2011, 16(5-6):573-600.
44. Vinti G., Zampogni L. A unified approach for the convergence of linear Kantorovich-type operators. Adv. Nonlinear Stud. 2014, in press.