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Time Operators and Approximation of Continuous Functions

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Abstract

The main purpose of this paper is to study time operators associated with generalized shifts and determined by the Haar and Faber–Schauder bases on the space of continuous functions. It is given the characterization of the domains of the constructed time operators and their scalings. It is also shown how scalings of time operators affect the dynamics of associated semigroups of shift operators.

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References

  1. Antoniou, I., Suchanecki, Z.: Nonuniform time operator, chaos and wavelets on the interval. Chaos Solitons Fractals 11, 423–435 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoniou, I., Suchanecki, Z.: Time operators, innovations and approximations. Chaos Solitons Fractals 17, 337–342 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antoniou, I., Sadovnichii, V.A., Shkarin, S.A.: Time operators and shift representation of dynamical systems. Physica A 269, 299–313 (1999)

    Article  Google Scholar 

  4. Ciesielski, Z.: On Haar functions and the Schauder basis of the space C 〈0,1〉. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 7, 227–232 (1959)

    MathSciNet  MATH  Google Scholar 

  5. Ciesielski, Z.: Properties of the orthonormal Franklin system. Stud. Math. 23, 141–157 (1963)

    MathSciNet  MATH  Google Scholar 

  6. Ciesielski, Z.: Properties of the orthonormal Franklin system II. Stud. Math. 27, 289–323 (1966)

    MathSciNet  MATH  Google Scholar 

  7. Ciesielski, Z., Kwapień, S.: Some properties of the Haar, Walsh–Paley, Franklin and the bounded polygonal orthonormal bases in L p spaces. Comment. Math. Spec. vol. II, Dedic. L. Orlicz, 37–42 (1979)

  8. Goldstein, S., Misra, B., Courbage, M.: On intrinsic randomness of dynamical systems. J. Stat. Phys. 25, 111–126 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Gómez, F.: Nonunitary similarity transformation of conservative to dissipative evolutions: Intertwining without time operator. J. Math. Phys. 48, 043506 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  10. Kaczmarz, S., Steinhaus, H.: Theorie der Orthogonalreihen. Monografje Matematyczne, Warszawa (1935)

    Google Scholar 

  11. Lasota, A., Mackey, M.: Chaos, Fractals, and Noise. Springer, New York (1994)

    MATH  Google Scholar 

  12. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I and II. Springer, Berlin (1977)

    Google Scholar 

  13. Misra, B.: Nonequilibrium entropy, Lyapunov variables, and ergodic properties of classical systems. Proc. Natl. Acad. Sci. USA 75, 1627–1631 (1978)

    Article  ADS  Google Scholar 

  14. Misra, B., Prigogine, I., Courbage, M.: From deterministic dynamics to probabilistic descriptions. Physica A 98, 1–26 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  15. Prigogine, I.: From Being to Becoming. Freeman, New York (1980)

    Google Scholar 

  16. Suchanecki, Z.: An L 1 extension of stochastic dynamics for irreversible systems. In: Lecture Notes in Mathematics, vol. 1391, pp. 367–374. Springer, Berlin (1989)

  17. Suchanecki, Z.: On lambda and internal time operators. Physica A 187, 249–266 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  18. Suchanecki, Z., Antoniou, I.: Time operators, innovations and approximations. Chaos Solitons Fractals 17, 337–342 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suchanecki, Z.: An explicit form of the time operator on the Fock space of a gaussian stationary process. Int. J. Theor. Phys. 43, 1097–1109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schipp, F., Wade, W.R., Simon, P., Pal, J.: Walsh Series—an Introduction to Dyadic Harmonic Analysis. Hilger, Bristol (1990)

    MATH  Google Scholar 

  21. Zygmund, A.: Trigonometric Series, vols. I, II. Cambridge University Press, Cambridge (1968)

    Google Scholar 

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Authors and Affiliations

  1. Université du Luxembourg, Campus Kirchberg, Rue Coudenhove-Kalergi 6, 1359, Luxembourg, Luxembourg

    U. Sorger & Z. Suchanecki

  2. Institute Mathematics and Informatics, University of Opole, 45-052, Opole, Poland

    Z. Suchanecki

Authors
  1. U. Sorger
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  2. Z. Suchanecki
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Correspondence to Z. Suchanecki.

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Sorger, U., Suchanecki, Z. Time Operators and Approximation of Continuous Functions. Int J Theor Phys 47, 1792–1808 (2008). https://doi.org/10.1007/s10773-007-9622-9

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  • DOI: https://doi.org/10.1007/s10773-007-9622-9

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