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The Dirichlet ring and unconditional bases in L 2[0, 2π]


It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.

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  1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1976.

    Book  MATH  Google Scholar 

  2. K. I. Babenko, “On conjugate functions,” Dokl. Akad. Nauk SSSR, 62 (1948), 157–160.

    MathSciNet  MATH  Google Scholar 

  3. A. Beurling, “The collected works of Arno Beurling,” in: Harmonic Analysis, Contemp. Math., vol. 2, Birkhäuser, Boston, 1989, 378–380.

    Google Scholar 

  4. L. Carleson, “On convergence and growth of partial sums of Fourier series,” Acta Math., 116 (1966), 135–157.

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Chandrasekharan, Arithmetical Functions, Springer-Verlag, New York-Heidelberg-Berlin, 1970.

    Book  MATH  Google Scholar 

  6. P. Djakov and B. Mityagin, “Bari-Markus property for Riesz projections of 1D periodic Dirac operators,” Math. Nachr., 283 (2010), 443–462.

    Article  MathSciNet  MATH  Google Scholar 

  7. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Math. Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969.

    MATH  Google Scholar 

  8. H. Hedenmalm, P. Lindqvist, and K. Seip, “A Hilbert space of Dirichlet series and systems of dilated functions in L 2(0, 1),” Duke Math. J., 86 (1997), 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Kato, Perturbation Theory for Linear Operators; Corr. Printing of the 2nd Ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980.

    Google Scholar 

  10. A. M. Olevskii, “On operators generating conditional bases in a Hilbert space,” Mat. Zametki, 12:1 (1972), 73–84; English transl.: Math. Notes, 12:1 (1972), 476–482.

    MathSciNet  MATH  Google Scholar 

  11. A. A. Shkalikov, “On the basis problem of eigenfunctions of an ordinary differential operator,” Uspekhi Matem. Nauk, 34:5(209) (1979), 235–236; English transl.: Russian Math. Surveys, 34:5 (1979), 249–250.

    MathSciNet  MATH  Google Scholar 

  12. A. Sowa, “A fast-transform basis with hysteretic features,” in: IEEE Conference Proceedings: Electrical and Computer Engineering (CCECE), 2011 24th Canadian Conference on, 8–11 May 2011, 000253–000257.

  13. A. Sowa, “Factorizing matrices by Dirichlet multiplication,” Linear Algebra Appl., 438:5 (2013), 2385–2393.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Sowa, “On an eigenvalue problem with a reciprocal-linear term,” Waves in Random and Complex Media, 22:2 (2012), 186–206.

    Article  MathSciNet  Google Scholar 

  15. R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York-London-Toronto, 1980.

    MATH  Google Scholar 

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Correspondence to Artur Sowa.

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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 47, No. 3, pp. 75–81, 2013

Original Russian Text Copyright © by Artur Sowa

The author gratefully acknowledges the support of the Canadian Foundation for Innovation, grant LOF no. 22117.

Since all bounded unconditional bases are Riesz, we will henceforth use the two terms interchangeably.

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Sowa, A. The Dirichlet ring and unconditional bases in L 2[0, 2π]. Funct Anal Its Appl 47, 227–232 (2013).

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