Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 921–932 | Cite as

An Affirmative Result of the Open Question on Determining Function Jumps by Spline Wavelets

  • Hai Ying Zhang
  • Xian Liang Shi
  • Jian Zhong Wang
Article
  • 13 Downloads

Abstract

We study the open question on determination of jumps for functions raised by Shi and Hu in 2009. An affirmative answer is given for the case that spline-wavelet series are used to approximate the functions.

Keywords

Jump B-spline wavelets 

MR(2010) Subject Classification

42A50 42A16 

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Notes

Acknowledgements

The authors thank the referees for helpful suggestions and their time.

References

  1. [1]
    Banerjee, N. S., Geer, J.: Exponentially accurate approximations to piecewise smooth periodic functions, ICASE report 95–17, NASA Langley Research Center, 1955Google Scholar
  2. [2]
    Chen, Y., Shi, X. L.: Determination of jumps in terms of derivative convolution operators. Acta Math. Hungar., 134(4), 372–392 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Chui, C. K.: An Introduction to Wavelets, Academic Press, Inc., Boston, 1992MATHGoogle Scholar
  4. [4]
    Chui, C. K., Shi, X. L.: On L p-boundedness of affine frame operators. Indag. Math. (N.S.), 4(4), 431–438 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Chui, C. K., Wang, J. Z.: On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc., 330, 903–915 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Devore, R. A., Lorentz, G. G.: Constructive Approximation, Springer, Berlin, 2010MATHGoogle Scholar
  7. [7]
    Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal., 7, 101–135 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Golubov, B. I.: Determination of jump of function of bounded variation by its Fourier series. Math. Notes, 12, 444–449 (1975)CrossRefMATHGoogle Scholar
  9. [9]
    Hu, L., Shi, X. L.: Concentration factors for functions with harmonic bounded mean variation. Acta Math. Hungar., 116(1–2), 89–103 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Kelly, S. E., Kon, M. A., Rephal, L. A.: Local convergence for wavelet expansions. J. Funct. Anal., 26, 102–138 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Kvernadge, G.: Determination of jumps of a bounded function by its Fourier Series. J. Approx. Theory, 92, 167–190 (1998)MathSciNetCrossRefGoogle Scholar
  12. [12]
    Lukács, F.: über die Bestimmung des sprunges einer Funktion aus ihrer Fourierreihe. J. Reine Angew. Math., 150, 107–112 (1920)MathSciNetMATHGoogle Scholar
  13. [13]
    Moricz, F.: Determination of jumps in terms of Abel Poisson means. Acta. Math. Hungar., 98(3), 259–262 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Moricz, F.: Ferenc Lukács type theorem in terms of Abel-Possion means of conjugate series. Proc. Amer. Math. Soc., 131, 1243–1250 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Shi, Q. L., Shi, X. L.: Determination of jumps in terms of spectral data. Acta. Math. Hungar., 110(3), 193–206 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Shi, X. L., Hu, L.: Determination of jumps for functions based on Malvar–Coifman–Meyer conjugate wavelets. Sci. China, Ser. A, 52(3), 443–456 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Shi, X. L., Wang, W.: Applications of operators to determination of jumps for functions. Acta Math. Hungar., 134(4), 439–451 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Shi, X. L., Zhang, H. Y.: Determination of jumps via advanced concentration factors. Appl. Comput. Harmon. Anal., 26, 1–13 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Shi, X. L., Zhang, H. Y.: Improvement of convergence rate for the Moricz process. Acta Sci Math. (Szeged) 76(3–4), 471–486 (2010)MathSciNetMATHGoogle Scholar
  20. [20]
    Zygmund, A.: Trigonometric Series, Cambridge University Press, Cambridge UK, 1959MATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Hai Ying Zhang
    • 1
  • Xian Liang Shi
    • 2
  • Jian Zhong Wang
    • 3
  1. 1.Science of CollegeHangzhou Dianzi UniversityHangzhouP. R. China
  2. 2.College of Mathematics and Computer ScienceHu’nan Normal UniversityChangshaP. R. China
  3. 3.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA

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