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Generating operator for discrete Chrestenson functions

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Abstract

We study the structure and properties of and a construction algorithm for a cyclic operator whose orbit consists of discrete Chrestenson functions arranged as columns of the matrix of a Kronecker power of the discrete Fourier transform. We analyze possibilities for extending the results to the case of discrete Vilenkin functions.

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References

  1. Trakhtman, A.M. and Trakhtman, V.A., Osnovy teorii diskretnykh signalov na konechnykh intervalakh (Elements of Theory of Discrete Signals on Finite Intervals), Moscow: Sov. Radio, 1975.

    Google Scholar 

  2. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.

    Book  MATH  Google Scholar 

  3. Malozemov, V.N. and Masharskii, S.M., Generalized Wavelet Bases Associated with the Discrete Vilenkin-Chrestenson Transformation, Algebra i Analiz, 2001, vol. 13, no. 1, pp. 111–157 [St. Petersburg Math. J. (Engl. Transl.), 2002, vol. 13, no. 1, pp. 75–106].

    MathSciNet  Google Scholar 

  4. Golubov, B.I., Efimov, A.B., and Skvortsov, V.A., Ryady i preobrazovaniya Uolsha: teoriya i primeneniya, Moscow: Nauka, 1987. Translated under the title Walsh Series and Transforms: Theory and Applications, Dordrecht: Kluwer, 1991.

    MATH  Google Scholar 

  5. Bespalov, M.S., The Discrete Chrestenson Transform, Probl. Peredachi Inf., 2010, vol. 46, no. 4, pp. 91–115 [Probl. Inf. Trans. (Engl. Transl.), 2010, vol. 46, no. 4, pp. 353–375].

    MathSciNet  Google Scholar 

  6. Bespalov, M.S., Discrete Chrestenson Functions, Izbrannye glavy diskretnogo garmonicheskogo analiza i geometricheskogo modelirovaniya (Selected Chapters in Discrete Harmonic Analysis and Computer Aided Geometric Design), Malozemov, V.N., Ed., Part 2, St. Petersburg: St. Petersburg State Univ., 2014, pp. 513–521.

    Google Scholar 

  7. Bespalov, M.S., Eigenspaces of the Discrete Walsh Transform, Probl. Peredachi Inf., 2010, vol. 46, no. 3, pp. 60–79 [Probl. Inf. Trans. (Engl. Transl.), 2010, vol. 46, no. 3, pp. 253–271].

    MathSciNet  Google Scholar 

  8. Bespalov, M.S., Sequence of Operators Generating DiscreteWalsh Functions, Izbrannye glavy diskretnogo garmonicheskogo analiza i geometricheskogo modelirovaniya (Selected Chapters in Discrete Harmonic Analysis and Computer Aided Geometric Design), Malozemov, V.N., Ed., Part 2, St. Petersburg: St. Petersburg State Univ., 2014, pp. 595–602.

    Google Scholar 

  9. Malozemov, V.N. and Masharskii, S.M., Osnovy diskretnogo garmonicheskogo analiza (Basics of Discrete Harmonic Analysis), St. Petersburg: Smirnov Scientific Research Institute of Mathematics and Mechanics, 2003.

    Google Scholar 

  10. Bespalov, M.S., On the Properties of a New Tensor Product of Matrices, Zh. Vychisl. Mat. Mat. Fiz., 2014, vol. 54, no. 4, pp. 547–561 [Comput. Math. Math. Phys. (Engl. Transl.), 2014, vol. 54, no. 4, pp. 561–574].

    MATH  MathSciNet  Google Scholar 

  11. Marcus, M. and Minc, H., A Survey of Matrix Theory and Matrix Inequalities, Boston: Allyn and Bacon, 1964. Translated under the title Obzor po teorii matrits i matrichnykh neravenstv, Moscow: Nauka, 1972.

    MATH  Google Scholar 

  12. Halmos, P.R., Finite-Dimensional Vector Spaces, Princeton, N.J.: Van Nostrand, 1958, 2nd ed. Translated under the title Konechnomernye vektornye prostranstva, Moscow: Nauka, 1963.

    MATH  Google Scholar 

  13. Hall, M., Jr., Combinatorial Theory, Waltham: Blaisdell, 1967. Translated under the title Kombinatorika, Moscow: Mir, 1970.

    MATH  Google Scholar 

  14. Akhiezer, N.I. and Glazman, I.M., Teoriya lineinykh operatorov v gil’bertovom prostranstve, Moscow: Nauka, 1966. Translated under the title Theory of Linear Operators in Hilbert Space, Boston: Pitman, 1981.

    Google Scholar 

  15. Gantmakher, F.R., Teoriya matrits, Moscow: Nauka, 1966, 2nd ed. Translated under the title The Theory of Matrices, 2 vols., Providence: AMS Chelsea, 1998.

    Google Scholar 

  16. Schipp, F., Wade, W.R., Simon, P., and Pal, J., Walsh Series. An Introduction to Dyadic Harmonic Analysis, Budapest: Acad. Kiadó, 1990.

    MATH  Google Scholar 

  17. Vilenkin, N.Ya., On a Class of Complete Orthonormal Systems, Izv. Akad. Nauk SSSR, Ser. Mat., 1947, vol. 11, no. 4, pp. 363–400.

    MATH  MathSciNet  Google Scholar 

  18. Bespalov, M.S., Discrete Vilenkin Functions, Izbrannye glavy diskretnogo garmonicheskogo analiza i geometricheskogo modelirovaniya (Selected Chapters in Discrete Harmonic Analysis and Computer Aided Geometric Design), Malozemov, V.N., Ed., Part 2, St. Petersburg: St. Petersburg State Univ., 2014, pp. 553–563.

    Google Scholar 

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Authors and Affiliations

  1. Vladimir State University, Vladimir, Russia

    M. S. Bespalov

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  1. M. S. Bespalov
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Correspondence to M. S. Bespalov.

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Original Russian Text © M.S. Bespalov, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 1, pp. 42–53.

The research is made under the State Task no. 2014/13 of the Ministry of Education and Science of the Russian Federation, project code 3037.

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Bespalov, M.S. Generating operator for discrete Chrestenson functions. Probl Inf Transm 51, 37–48 (2015). https://doi.org/10.1134/S0032946015010056

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  • DOI: https://doi.org/10.1134/S0032946015010056

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