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Efficiency of weak greedy algorithms for m-term approximations

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Abstract

We investigate the efficiency of weak greedy algorithms for m-term expansional approximation with respect to quasi-greedy bases in general Banach spaces. We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm (WTGA) and weak Chebyshev thresholding greedy algorithm. Then we discuss the greedy approximation on some function classes. For some sparse classes induced by uniformly bounded quasi-greedy bases of L p , 1 < p < ∞, we show that the WTGA realizes the order of the best m-term approximation. Finally, we compare the efficiency of the weak Chebyshev greedy algorithm (WCGA) with the thresholding greedy algorithm (TGA) when applying them to quasi-greedy bases in L p , 1 ≤ p < ∞, by establishing the corresponding Lebesgue-type inequalities. It seems that when p > 2 the WCGA is better than the TGA.

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References

  1. Dan W. Analysis of orthogonal multi-matching pursuit under restricted isometry property. Sci China Math, 2014, 57: 2179–2188

    Article  MathSciNet  MATH  Google Scholar 

  2. Dan W, Wang R. Robustness of orthogonal matching pursuit under restricted isometry property. Sci China Math, 2014, 57: 627–634

    Article  MathSciNet  MATH  Google Scholar 

  3. DeVore R A. Nonlinear approximation. Acta Numer, 1998, 7: 51–150

    Article  MathSciNet  MATH  Google Scholar 

  4. DeVore R A, Temlyakov V N. Nonlinear approximation by trigonometric sums. J Fourier Anal Appl, 1995, 2: 29–48

    Article  MathSciNet  MATH  Google Scholar 

  5. DeVore R A, Temlyakov V N. Some remarks on greedy algorithms. Adv Comput Math, 1996, 5: 173–187

    Article  MathSciNet  MATH  Google Scholar 

  6. Dilworth S J, Kalton N J, Kutzarova D. On the existence of almost greedy bases in Banach spaces. Studia Math, 2003, 159: 67–101

    Article  MathSciNet  MATH  Google Scholar 

  7. Dilworth S J, Kalton N J, Kutzarova D, et al. The thresholding greedy algorithm, greedy bases, and duality. Constr Approx, 2003, 19: 575–597

    Article  MathSciNet  MATH  Google Scholar 

  8. Dilworth S J, Kutzarova D, Oikhberg T. Lebesgue constants for the weak greedy algorithm. Rev Mat Complut, 2015, 28: 393–409

    Article  MathSciNet  MATH  Google Scholar 

  9. Dilworth S J, Kutzarova D, Temlyakov V N. Convergence of some greedy algorithms in Banach spaces. J Fourier Anal Appl, 2002, 8: 489–506

    Article  MathSciNet  MATH  Google Scholar 

  10. Dilworth S J, Soto-Bajo M, Temlyakov V N. Quasi-greedy bases and Lebesgue-type inequalities. Studia Math, 2012, 211: 41–69

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu X, Lin H B, Yang D C, et al. Hardy spaces H p over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58: 309–388

    Article  MathSciNet  MATH  Google Scholar 

  12. Garrigos G, Hernandez E, Oikhberg T. Lebesgue-type inequalities for quasi-greedy bases. Constr Approx, 2013, 38: 447–470

    Article  MathSciNet  MATH  Google Scholar 

  13. Hajek P, Montesinos Santalucia V, Vanderwerff J, et al. Biorthogonal Systems in Banach Spaces. New York: Springer, 2008

    MATH  Google Scholar 

  14. Kahane J P. Some Random Series of Functions. Cambridge: Cambridge University Press, 1993

    MATH  Google Scholar 

  15. Konyagin S V, Temlyakov V N. A remark on greedy approximation in Banach spaces. East J Approx, 1999, 5: 365–379

    MathSciNet  MATH  Google Scholar 

  16. Konyagin S V, Temlyakov V N. Greedy approximation with regard to bases and general minimal systems. Serdica Math J, 2002, 28: 305–328

    MathSciNet  MATH  Google Scholar 

  17. Lin H B, Yang D C. Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci China Math, 2014, 57: 123–144

    Article  MathSciNet  MATH  Google Scholar 

  18. Lindenstrauss J, Tzafriri L. Classical Banach Spaces II. Berlin: Springer, 1979

    Book  MATH  Google Scholar 

  19. Long J F, Ye P X. Weak greedy algorithms for nonlinear approximation with quasi-greedy bases. WSEAS Trans Math, 2014, 13: 525–534

    Google Scholar 

  20. Nielsen M. An example of an almost greedy uniformly bounded orthonormal basis for L p(0, 1). J Approx Theory, 2007, 149: 188–192

    Article  MathSciNet  MATH  Google Scholar 

  21. Temlyakov V N. Greedy algorithm and m-term trigonometric approximation. Constr Approx, 1998, 14: 569–587

    Article  MathSciNet  MATH  Google Scholar 

  22. Temlyakov V N. Greedy algorithms in Banach spaces. Adv Comput Math, 2001, 14: 277–292

    Article  MathSciNet  MATH  Google Scholar 

  23. Temlyakov V N. Nonlinear methods of approximation. Found Comput Math, 2003, 3: 33–107

    Article  MathSciNet  MATH  Google Scholar 

  24. Temlyakov V N. Greedy approximation. Acta Numer, 2008, 17: 235–409

    Article  MathSciNet  MATH  Google Scholar 

  25. Temlyakov V N. Sparse approximation and recovery by greedy algorithms in Banach spaces. In: Forum of Mathematics, Sigma, vol. 2. Cambridge: Cambridge University Press, 2014, e12

    Google Scholar 

  26. Temlyakov V N, Yang M R, Ye P X. Greedy approximation with regard to non-greedy bases. Adv Comput Math, 2011, 34: 319–337

    Article  MathSciNet  MATH  Google Scholar 

  27. Temlyakov V N, Yang M R, Ye P X. Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. East J Approx, 2011, 17: 203–214

    MathSciNet  MATH  Google Scholar 

  28. Wang J X, Qian T. Approximation of monogenic functions by higher order Szegö kernels on the unit ball and half space. Sci China Math, 2014, 57: 1785–1797

    Article  MathSciNet  MATH  Google Scholar 

  29. Wojtaszczyk P. Greedy algorithm for general biorthogonal systems. J Approx Theory, 2000, 107: 293–314

    Article  MathSciNet  MATH  Google Scholar 

  30. Yuan W, Sickel W, Yang D C. Interpolation of Morrey-Campanato and related smoothness spaces. Sci China Math, 2015, 58: 1835–1908

    Article  MathSciNet  MATH  Google Scholar 

  31. Zygmund A. Trigonometric Series. Cambridge: Cambridge University Press, 1959

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    PeiXin Ye & XiuJie Wei

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  1. PeiXin Ye
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  2. XiuJie Wei
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Correspondence to XiuJie Wei.

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Ye, P., Wei, X. Efficiency of weak greedy algorithms for m-term approximations. Sci. China Math. 59, 697–714 (2016). https://doi.org/10.1007/s11425-015-5106-1

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