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Variational measures related to local systems and the Ward property of \(\mathcal{P}\)-adic path bases

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Abstract

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a \(\mathcal{P}\)-adic path system that defines a differentiation basis which does not possess Ward property.

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References

  1. G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli and A. I. Rubinshtein: Multiplicative systems of functions and harmonic analysis on zero-dimensional groups. Baku (1981). (In Russian.)

  2. E. S. Bajgogin: On a dyadic Perron integral. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 48 (1993), 25–28.

    Google Scholar 

  3. B. Bongiorno, L. Di Piazza and D. Preiss: Infinite variation and derivatives in ℝn. J. Math. Anal. Appl. 224 (1998), 22–33.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Bongiorno, L. Di Piazza and V. A. Skvortsov: A new full descriptive characterization of the Denjoy-Perron integral. Real Anal. Exchange. 21 (1995–96), 656–663.

    MATH  MathSciNet  Google Scholar 

  5. B. Bongiorno, L. Di Piazza and V. A. Skvortsov: On variational measures related to some bases. J. Math. Anal. Appl. 250 (2000), 533–547.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Bongiorno, L. Di Piazza and V. A. Skvortsov: On dyadic integrals and some other integrals associated with local systems. J. Math. Anal. Appl. 271 (2002), 506–524.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Bongiorno, L. Di Piazza and V. A. Skvortsov: The Ward property for a \(\mathcal{P}\)-adic basis and the \(\mathcal{P}\)-adic integral. J. Math. Anal. Appl. 285 (2003), 578–592.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Bongiorno, W. Pfeffer and B. S. Thomson: A full descriptive definition of the gage integral. Canad. Math. Bull. 39 (1996), 390–401.

    MATH  MathSciNet  Google Scholar 

  9. Z. Buczolich and W. Pfeffer: When absolutely continuous implies σ-finite. Acad. Roy. Belg. Bull. Cl. Sci. 8 (1997), 155–160.

    MathSciNet  Google Scholar 

  10. Z. Buczolich and W. Pfeffer: On absolute continuity. J. Math. Anal. Appl. 222 (1998), 64–78.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Cai-shi and D. Chuan-Song: An integral involving Thomson’s local systems. Real Anal. Exchange 19 (1993/94), 248–253.

    MathSciNet  Google Scholar 

  12. L. Di Piazza: Variational measures in the theory of the integration in R m. Czechoslovak Math. J. 51 (2001), 95–110.

    Article  MATH  MathSciNet  Google Scholar 

  13. V. Ene: Real functions-Current Topics. Lecture Notes in Math., Vol. 1603, Springer-Verlag, 1995.

  14. V. Ene: Thomson’s variational measures. Real Anal. Exchange 24 (1998/99), 523–565.

    MATH  MathSciNet  Google Scholar 

  15. Sh. Fu: Path integral: an inversion of path derivatives. Real Anal. Exchange 20 (1994–95), 340–346.

    MathSciNet  Google Scholar 

  16. B. Golubov, A. Efimov and V. A. Skvortsov: Walsh series and transforms-Theory and applications. Kluwer Academic Publishers, 1991.

  17. R. A. Gordon: The inversion of approximate and dyadic derivatives using an extension of the Henstock integral. Real Anal. Exchange 16 (1990–91), 154–168.

    MathSciNet  Google Scholar 

  18. R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics., Vol. 4, AMS, Providence, 1994.

    MATH  Google Scholar 

  19. J. Jarník and J. Kurzweil: Perron-type integration on n-dimensional intervals and its properties. Czechoslovak Math. J. 45 (1995), 79–106.

    MATH  MathSciNet  Google Scholar 

  20. J. Kurzweil and J. Jarník: Differentiability and integrability in n-dimension with respect to α-regular intervals. Results Math. 21 (1992), 138–151.

    MATH  MathSciNet  Google Scholar 

  21. Tuo-Yeong Lee: A full descriptive definition of the Henstok-Kurzwail integral in the Euclidean space. Proceedings of London Math. Society 87 (2003), 677–700.

    Article  MATH  Google Scholar 

  22. K. M. Ostaszewski: Henstock integration in the plane. Memoirs of the AMS, Providence Vol. 353, 1986.

  23. W. F. Pfeffer: The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge, 1993.

    MATH  Google Scholar 

  24. W. F. Pfeffer: The Lebesgue and Denjoy-Perron integrals from a descriptive point of view. Ricerche di Matematica Vol. 48, 1999.

  25. S. Saks: Theory of the integral. Dover, New York, 1964.

    MATH  Google Scholar 

  26. V. A. Skvortsov: Variation and variational measures in integration theory and some applications. J. Math. Sci. (New York) 91 (1998), 3293–3322.

    Article  MATH  MathSciNet  Google Scholar 

  27. V. A. Skvortsov and M. P. Koroleva: Series in multiplicative systems convergent to Denjoy-integrable functions. Mat. sb. 186 (1995), 129–150; Engl.transl. Math. USSR Sb. 186 (1995), 1821–1842.

    MATH  MathSciNet  Google Scholar 

  28. V. A. Skvortsov and F. Tulone: Generalized Henstock integrals in the theory of series with respect to multiplicative system. Vestnik Moskov. Gos. Univ. Ser. Mat. Mekh. 4 (2004), 7–11; engl. transl. Moscow Univ. Math. Bull. 59 (2004), 6–10.

    MathSciNet  Google Scholar 

  29. V. A. Skvortsov and Yu. A. Zherebyov: On classes of functions generating absolutely continuous variational measures. Real. Anal. Exchange 30 (2004–2005), 361–372.

    MATH  MathSciNet  Google Scholar 

  30. V. A. Skvortsov and Yu. A. Zherebyov: On Radon-Nikodim derivative for the variational measure constructed by dyadic basis. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2004), 6–12 (Engl. transl. Moscow Univ. Math. Bull. 59 (2004), 5–11).

  31. B. S. Thomson: Real functions. Lecture Notes in Math., Vol. 1170, Springer-Verlag. 1985.

  32. B. S. Thomson: Derivation bases on the real line. Real Anal. Exchange 8 (1982/83), 67–207 and 278–442.

    MathSciNet  Google Scholar 

  33. B. S. Thomson: Some property of variational measures. Real Anal. Exchange 24 (1998/99), 845–853.

    MATH  MathSciNet  Google Scholar 

  34. F. Tulone: On the Ward Theorem for \(\mathcal{P}\)-adic path bases associated with a bounded sequence \(\mathcal{P}\). Math. Bohem. 129 (2004), 313–323.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Metodi e Mod. Mat., Viale delle Scienye, 90100, Palermo, Italy

    D. Bongiorno

  2. Dipartimento di Matem. ed Applic., Via Archirafi 34, 90123, Palermo, Italy

    L. Di Piazza

  3. Department of Mathematics, Moscow State University, Moscow, 119899, Russia

    V. A. Skvortsov

  4. Instytut Matematyki, Uniwersytet Kazimierza Wielkego, 85-072, Bydgoszcz, Poland

    V. A. Skvortsov

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  1. D. Bongiorno
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  2. L. Di Piazza
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  3. V. A. Skvortsov
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Bongiorno, D., Di Piazza, L. & Skvortsov, V.A. Variational measures related to local systems and the Ward property of \(\mathcal{P}\)-adic path bases. Czech Math J 56, 559–578 (2006). https://doi.org/10.1007/s10587-006-0037-1

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  • DOI: https://doi.org/10.1007/s10587-006-0037-1

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