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


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.


Jump B-spline wavelets 

MR(2010) Subject Classification

42A50 42A16 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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


  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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Chui, C. K.: An Introduction to Wavelets, Academic Press, Inc., Boston, 1992zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Devore, R. A., Lorentz, G. G.: Constructive Approximation, Springer, Berlin, 2010zbMATHGoogle Scholar
  7. [7]
    Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal., 7, 101–135 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Golubov, B. I.: Determination of jump of function of bounded variation by its Fourier series. Math. Notes, 12, 444–449 (1975)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Kelly, S. E., Kon, M. A., Rephal, L. A.: Local convergence for wavelet expansions. J. Funct. Anal., 26, 102–138 (1994)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  13. [13]
    Moricz, F.: Determination of jumps in terms of Abel Poisson means. Acta. Math. Hungar., 98(3), 259–262 (2003)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Shi, X. L., Zhang, H. Y.: Determination of jumps via advanced concentration factors. Appl. Comput. Harmon. Anal., 26, 1–13 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  20. [20]
    Zygmund, A.: Trigonometric Series, Cambridge University Press, Cambridge UK, 1959zbMATHGoogle 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

Personalised recommendations