Home Lp Spaces in vector lattices and applications
Article
Licensed
Unlicensed Requires Authentication

Lp Spaces in vector lattices and applications

  • Antonio Boccuto EMAIL logo , Domenico Candeloro and Anna Rita Sambucini
Published/Copyright: November 30, 2017
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 6

Abstract

Lp spaces are investigated for vector lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the vector lattice context, and some properties of the Brownian motion and the Brownian bridge are studied, to solve some stochastic differential equations.

MSC 2010: 28B15; 41A35; 46G10
Keywords: vector lattice; filter convergence; modular; Lp space; Hermite-Hadamard inequality; Schwartz inequality; Jensen inequality; Brownian motion

This work was supported by University of Perugia - Department of Mathematics and Computer Sciences - Grant Nr 2010.011.0403 and by the Grant prot. UFMBAZ2017/0000326 of GNAMPA - INDAM (Italy).

A. Boccuto orcid id: 0000-0003-3795-8856, D. Candeloro orcid id: 0000-0003-0526-5334, A. R. Sambucini orcid id: 0000-0003-0161-8729.

Dedicated to Professor Paolo de Lucia on the occasion of his 80th birthday

Communicated by Anatolij Dvurečenskij

References

[1] Angeloni, L.—Vinti, G.: Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators, Numer. Funct. Anal. Optim. 31 (2010), 519–548.10.1080/01630563.2010.490549Search in Google Scholar

[2] Angeloni, L.—Vinti, G.: Approximation in variation by homothetic operators in multidimensional setting, Differential Integral Equations 26 (2013), 655–674.10.57262/die/1363266083Search in Google Scholar

[3] Bardaro, C.—Boccuto, A.—Dimitriou, X.—Mantellini, I.: Modular filter convergence theorems for abstract sampling-type operators, Appl. Anal. 92 (2013), 2404–2423.10.1080/00036811.2012.738480Search in Google Scholar

[4] 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. 316 (2006), 269–306.10.1016/j.jmaa.2005.04.042Search in Google Scholar

[5] Bardaro, C.—Butzer, P. L.—Mantellini, I.: The foundations of fractional calculus in Mellin transform setting and applications, J. Fourier Anal. Appl. 21 (2015), 961–1017.10.1007/s00041-015-9392-3Search in Google Scholar

[6] Bardaro, C.—Butzer, P. L.—Mantellini, I.: The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics, Integral Transforms Spec. Funct. 27 (2016), 17–29.10.1080/10652469.2015.1087401Search in Google Scholar

[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. 6 (2006), 29–52.10.1007/BF03549462Search in Google Scholar

[8] 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 56 (2010), 614–633.10.1109/TIT.2009.2034793Search in Google Scholar

[9] Bardaro, C.—Musielak, J.—Vinti, G.: Nonlinear Integral Operators and Applications, de Gruyter, Berlin, 2003.10.1515/9783110199277Search in Google Scholar

[10] Billingsley, P.: Probability and Measure, John Wiley & Sons, New York, 1986.Search in Google Scholar

[11] Basile, A.—Bhaskara Rao, K. P. S.: Completeness of LP-spaces in the finitely additive setting and related stories, J. Math. Anal. Appl. 248 (2000), 588–624.10.1006/jmaa.2000.6946Search in Google Scholar

[12] Boccuto, A.—Candeloro, D.: Integral and differential calculus in Riesz spaces and applications, J. Appl. Funct. Anal. 3 (2008), 89–111.Search in Google Scholar

[13] Boccuto, A.—Candeloro, D.: Integral and ideals in Riesz spaces, Inform. Sci. 179 (2009), 647–660.10.1016/j.ins.2008.11.001Search in Google Scholar

[14] Boccuto, A.—Candeloro, D.: Differential calculus in Riesz spaces and applications to g-calculus, Mediterr. J. Math. 8 (2011), 315–329.10.1007/s00009-010-0072-xSearch in Google Scholar

[15] Boccuto, A.—Candeloro, D.—Sambucini, A. R.: Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes, J. Math. Anal. Appl. 419 (2014), 818–838. https://doi.org/10.1016/j.jmaa.2014.05.014.10.1016/j.jmaa.2014.05.014Search in Google Scholar

[16] Boccuto, A.—Candeloro, D.—Sambucini, A. R.: Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), 363–383. https://doi.org/10.4171/RLM/710.10.4171/RLM/710Search in Google Scholar

[17] Boccuto, A.—Dimitriou, X.: Modular filter convergence theorems for Urysohn integral operators and applications, Acta Math. Sin. (Engl. Ser.) 29 (2013), 1055–1066.10.1007/s10114-013-1443-6Search in Google Scholar

[18] Boccuto, A.—Dimitriou, X.: Modular convergence theorems for integral operators in the context of filter exhaustiveness and applications, Mediterr. J. Math. 10 (2013), 823–842. https://doi.org/10.1007/s00009-012-0199-z.10.1007/s00009-012-0199-zSearch in Google Scholar

[19] Boccuto, A.—Dimitriou, X.: Convergence Theorems for Lattice Group-Valued Measures, Bentham Science Publ., U. A. E., 2015.10.2174/97816810800931150101Search in Google Scholar

[20] Boccuto, A.—Dimitriou, X.—Papanastassiou, N.: Schur lemma and limit theorems in lattice groups with respect to filters, Math. Slovaca 62 (2012), 1145–1166.10.2478/s12175-012-0070-5Search in Google Scholar

[21] Boccuto, A.—Sambucini, A. R.: Comparison between different types of abstract integrals in Riesz spaces, Rend. Circ. Mat. Palermo 46 (1997), 255–278.10.1007/BF02977030Search in Google Scholar

[22] Boccuto, A.—Sambucini, A. R.: Addendum to: Comparison between different types of abstract integrals in Riesz Spaces, Rend. Circ. Mat. Palermo 49 (2000), 395–396.10.1007/BF02904245Search in Google Scholar

[23] Breiman, P.: Probability, Addison-Wesley, Reading, 1968.Search in Google Scholar

[24] Butzer, P. L.—Nessel, R. J.: Fourier Analysis and Approximation. Pure Appl. Math. 40, Academic Press, New York, London, 1971.10.1007/978-3-0348-7448-9Search in Google Scholar

[25] Candeloro, D.—Sambucini, A. R.: Filter convergence and decompositions for vector lattice-valued measures, Mediterr. J. Math. 12 (2015), 621–637. https://doi.org/10.1007/s00009-014-0431-0.10.1007/s00009-014-0431-0Search in Google Scholar

[26] Candeloro, D.—Sambucini, A. R.: Order-type Henstock and McShane integrals in Banach lattice setting. IEEE 12th International Symposium on Intelligent Systems and Informatics (SISY), September 11–13, 2014, Subotica, Serbia, 2014, pp. 55–59.10.1109/SISY.2014.6923557Search in Google Scholar

[27] Cavaliere, P.—de Lucia, P.—De Simone, A.: Functions determining locally solid topological Riesz spaces continuously embedded in L0, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), 151–160.10.4171/RLM/699Search in Google Scholar

[28] Choksi, J. R.: Vitali’s convergence theorem on term by term integration, L’ Enseignement Mathématique 47 (2001), 269–285.Search in Google Scholar

[29] Cluni, F.—Costarelli, D.—Minotti, A. M.—Vinti, G.: Enhancement of thermographic images as tool for structural analysis in earthquake engineering, NDT & E International 70 (2015), 60–72.10.1016/j.ndteint.2014.10.001Search in Google Scholar

[30] Costarelli, D.—Vinti, G.: Order of approximation for sampling Kantorovich operators, J. Integral Equations Appl. 26 (2014), 345–368.10.1216/JIE-2014-26-3-345Search in Google Scholar

[31] de Lucia, P.—Weber, H.: Completeness of function spaces, Ricerche Mat. 39 (1990), 81–97.Search in Google Scholar

[32] Filter, W.: Representations of Archimedean Riesz spaces a survey, Rocky Mountain J. Math. 24 (1994), 771–851. https://doi.org/10.1216/rmjm/1181072375.10.1216/rmjm/1181072375Search in Google Scholar

[33] Garsia, A. M.—Rodemich, E.—Rumsey Jr, H.: A real variable lemma and the continuity of paths of some gaussian processes, Indiana Univ. Math. J. 20 (1970/71), 565–578.10.1512/iumj.1971.20.20046Search in Google Scholar

[34] Grobler, J. J.: Jensen’s and martingale inequalities in Riesz spaces, Indagationes Mathematicae 25 (2014), 275–295.10.1016/j.indag.2013.02.003Search in Google Scholar

[35] Grobler, J. J.—Labuschagne, C. C. A.: The Ito integral for Brownian motion in vector lattices: Part 1, J. Math. Anal. Appl. 423 (2015), 797–819.10.1016/j.jmaa.2014.08.013Search in Google Scholar

[36] Grobler, J. J.—Labuschagne, C. C. A.: The Ito integral for Brownian motion in vector lattices: Part 2, J. Math. Anal. Appl. 423 (2015), 820–833.10.1016/j.jmaa.2014.09.063Search in Google Scholar

[37] Grobler, J. J.—Labuschagne, C. C. A.—Marraffa, V.: Quadratic variation of martingales in Riesz spaces J. Math. Anal. Appl. 410 (2014), 418–426.10.1016/j.jmaa.2013.08.037Search in Google Scholar

[38] Holmes, R. B.: Mathematical foundations of signal processing, SIAM Review 21 (1979), 361–388.10.1137/1021053Search in Google Scholar

[39] Kozlowski, W. M.: Modular Function Spaces. Pure Appl. Math., Marcel Dekker, New York, 1988.Search in Google Scholar

[40] Letta, G.: Martingales et Integration Stochastique, Quaderni S. N. S. Pisa, 1984.Search in Google Scholar

[41] Luxemburg, W. A. J.—Zaanen, A. C.: Riesz Spaces, I, North-Holland Publ. Co., Amsterdam, 1971.Search in Google Scholar

[42] Maligranda, L.: A simple proof of the Hölder and the Minkowski inequality, Amer. Math. Monthly 102 (1995), 256–259.10.2307/2975013Search in Google Scholar

[43] Musielak, J.: Orlicz Spaces and Modular spaces. Lecture Notes in Math. 1034, Springer-Verlag, New York, 1983.10.1007/BFb0072210Search in Google Scholar

[44] Nualart, D.: Fractional Brownian motion: Stochastic calculus and applications. In: Proc. Int. Congress Math., Madrid, Spain, (EMS) Vol.3-74, 2006, pp. 1541–562.10.4171/022-3/74Search in Google Scholar

[45] Rao, M. M.—Ren, Z. D.: Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.Search in Google Scholar

[46] Stoica, G.: On some stochastic-type operators, An. Univ. Bucuresti Mat. 39, (1990), 58–62.Search in Google Scholar

[47] Stoica, G.: Martingales in vector lattices I and II, Soc. Sci. Roumanie 34 (1990), 357–362, and 35 (1991), 155–158.Search in Google Scholar

[48] Stoica, G.: Order convergence and decompositions of stochastic processes, An. Univ. Bucuresti Mat. 42 (1993), 85–91.Search in Google Scholar

[49] Vardy, J.—Watson, B. A.: On the decompositions of T-quasi-martingales on Riesz spaces, Positivity 18 (2014), 425–437.10.1007/s11117-013-0253-5Search in Google Scholar

[50] Vinti, G.—Zampogni, L.: Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theory 161 (2009), 511–528.10.1016/j.jat.2008.11.011Search in Google Scholar

[51] Vinti, G.—Zampogni, L.: A unifying approach to convergence of linear sampling type operators in Orlicz spaces, Adv. Diff. Equations 16 (2011), 573–600.10.57262/ade/1355703301Search in Google Scholar

[52] Wright, J. D. M.: The measure extension problem for vector lattices, Ann. Inst. Fourier, Grenoble 21 (1971), 65–85.10.5802/aif.393Search in Google Scholar

[53] Zaanen, A. C.: Riesz Spaces II, North-Holland Publ. Co., Amsterdam, 1983.Search in Google Scholar

Received: 2016-3-16
Accepted: 2016-5-1
Published Online: 2017-11-30
Published in Print: 2017-11-27

© 2017 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
https://doi.org/10.1515/ms-2017-0060
Keywords for this article
vector lattice; filter convergence; modular; Lp space; Hermite-Hadamard inequality; Schwartz inequality; Jensen inequality; Brownian motion

Articles in the same Issue

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 18.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0060/html?lang=en
Scroll to top button