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Divergence of the greedy algorithm in the Faber-Schauder system on a continuum cardinality set

  • Fourier Analysis
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Abstract

The paper establishes that there exist a continuum cardinality set E 0 ⊂ [0, 1] and a function f 0(x) ∈ C [0,1], such that the greedy algorithm of f 0(x) with respect to the Faber-Schauder system converges to +∞ at all points of E 0.

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References

  1. V. N. Temlyakov, “Nonlinear Methods of Approximation,” Found. Comput. Math. 3, 33–107 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  2. V. N. Temlyakov, “Greedy Algorithm and m-term Trigonometric Approximation,” Constructive Approx. 14, 569–587 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  3. R. A. DeVore and V. N. Temlyakov, “Some Remarks on Greedy Algorithms,” Advances in Computational Math. 5, 173–187 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Wojtaszczyk, “Greedy Algorithm for General Biorthogonal Systems,” Journal of Approximation Theory 107, 293–314 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. V. Konyagin and V. N. Temlyakov, “A Remark on Greedy Approximation in Banach Spaces,” East J. on Approx. 5, 1–15 (1999).

    Google Scholar 

  6. T. W. Kőrner, “Decreasing Rearranged Fourier Series,” J. Fourier Anal. Appl. 5, 1–19 (1999).

    Article  MathSciNet  Google Scholar 

  7. M. G. Grigoryan, “On Convergence in L p Metric of Greedy Algorithm in Trigonometric System,” Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 39(5), 35–52 (2004).

    MathSciNet  Google Scholar 

  8. M. G. Grigoryan and S. L. Gogyan, “On Nonlinear Approximation with Respect to the Haar System and Modifications of Functions,” An. Math. 32, 49–80 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  9. M. G. Grigoryan and R. E. Zink, “Greedy Approximation with Respect to Certain Subsystems of the Walsh Orthonormal System,” Pros. Amer. Math. Soc. 134, 3495–3505 (2006).

    Article  MathSciNet  Google Scholar 

  10. S. J. Dilworth, N. J. Kalton and D. Kutzarova, “On the Existence of Almost Greedy Bases in Banach Spaces,” Studia Mathematica 159(1), 67–101 (2005).

    Article  MathSciNet  Google Scholar 

  11. S. Gogyan, “Greedy Algorithm with Regard to Haar Subsystems,” East J. Approx. 11, 221–236 (2005).

    MathSciNet  MATH  Google Scholar 

  12. A. A. Sargsyan, “The Quasigreedy System Property of some Subsystems of the Faber-Schauder System,” Izv. NAN Armenii, Matematika, [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 40(3), 46–54 (2005).

    MathSciNet  Google Scholar 

  13. A. A. Sargsian, “Greedy Algorithm with Regard to Schauder Subsystems,” in Proceedings of International Workshop on Harmonic Analysis and Approximations, Tsahkadzor, Armenia (2005), p. 65.

  14. A. A. Sargsian, “Quasigreedy and Democratic Subsystems of Faber-Schauder System,” Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 41(2), 56–73 (2006).

    Google Scholar 

  15. J. Schauder, “Zur Theorie Stetiger Abbildungen in Functionalraumen,” Math. Zeit. 26, 47–65 (1927).

    Article  MathSciNet  MATH  Google Scholar 

  16. G. M. Fikhtenholz, Course of Differential and Integral Calculus, Vol. 2 (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Yerevan State University, Armenia

    A. A. Sargsyan

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  1. A. A. Sargsyan
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Original Russian Text © A. A. Sargsyan, 2007, published in Izvestiya NAN Armenii. Matematika, 2007, No. 2, pp. 63–72.

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Sargsyan, A.A. Divergence of the greedy algorithm in the Faber-Schauder system on a continuum cardinality set. J. Contemp. Mathemat. Anal. 42, 109–115 (2007). https://doi.org/10.3103/S1068362307020070

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  • DOI: https://doi.org/10.3103/S1068362307020070

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