Vietnam Journal of Mathematics

, Volume 44, Issue 2, pp 419–429 | Cite as

A Study of the Sequence of Norm of Derivatives (or Primitives) of Functions Depending on Their Beurling Spectrum



In this paper, we characterize the behavior of the sequence of norm of derivatives (or primitives) of functions by their Beurling spectrum in Banach spaces. The Bernstein inequality for Banach spaces is also obtained.


Banach Spaces Beurling spectrum 

Mathematics Subject Classification (2010)

26D10 46E30 



This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.32. The authors would like to thank the referees for the exact corrections.


  1. 1.
    Abreu, L.D.: Real Paley–Wiener theorems for the Koornwinder–Swarttouw q-Hankel transform. J. Math. Anal. Appl. 334, 223–231 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Albrecht, E., Ricker, W.J.: Functional calculi and decomposability of unbounded multiplier operators in \(L^{p}(\mathbb R^{N})\). Proc. Edinb. Math. Soc. Ser. 2 38, 151–166 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Albrecht, E., Ricker, W.J.: Local spectral properties of certain matrix differential operators in \(L^{p}(\mathbb R^{N})^{m}\). J. Oper. Theory 35, 3–37 (1996)MathSciNetMATHGoogle Scholar
  4. 4.
    Andersen, N.B.: On real Paley–Wiener theorems for certain integral transforms. J. Math. Anal. Appl. 288, 124–135 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Andersen, N.B.: Real Paley–Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. Pac. J. Math. 213, 1–13 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Andersen, N.B.: Real Paley–Wiener theorems for the Hankel transform. J. Fourier Anal. Appl. 12, 17–25 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Andersen, N.B., De Jeu, M.: Elementary proofs of Paley–Wiener theorems for the Dunkl transform on the real line. Int. Math. Res. Not. 30, 1817–1831 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Andersen, N.B., De Jeu, M.: Real Paley-Wiener theorems and local spectral radius formulas. Trans. Am. Math. Soc. 362, 3613–3640 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bang, H.H.: A property of infinitely differentiable functions. Proc. Am. Math. Soc. 108, 73–76 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bang, H.H.: Functions with bounded spectrum. Trans. Am. Math. Soc. 347, 1067–1080 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bang, H.H.: A property of entire functions of exponential type. Analysis 15, 17–23 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bang, H.H.: The existence of a point spectral radius of pseudodifferential operators. Dokl. Math. 53, 420–422 (1996)MATHGoogle Scholar
  13. 13.
    Bang, H.H., Huy, V.N.: Behavior of the sequence of norm of primitives of a function. J. Approx. Theory 162, 1178–1186 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bang, H.H., Huy, V.N.: On the limit of norm of consecutive primitives of a function. East. J. Approx. 15, 111–122 (2009)MathSciNetMATHGoogle Scholar
  15. 15.
    Betancor, J.J., Betancor, J.D., Méndez, J.M.R.: Paley–Wiener type theorems for Chébli–Trimèche transforms. Publ. Math. Debr. 60, 347–358 (2002)MATHGoogle Scholar
  16. 16.
    Chettaoui, C., Trimèche, K.: New type Paley–Wiener theorems for the Dunkl transform on \(\mathbb {R}\). Integral Trans. Spec. Funct. 14, 97–115 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Giang, D.V.: Beurling spectrum of a function in a Banach space. Acta. Math. Vietnam 39, 305–312 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Garth Dales, H., Aeina, P., Eschmeier, J., Laursen, K., Willis, G.: Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press (2003)Google Scholar
  19. 19.
    De Jeu, M.: Some remarks on a proof of geometrical Paley–Wiener theorems for the Dunkl transform. Integral Trans. Spec. Funct. 18, 383–385 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    De Jeu, M.: Paley–Wiener theorems for the Dunkl transform. Trans. Am. Math. Soc. 358, 4225–4250 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Minh, N.V.: A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations. J. Differ. Equ. 247, 1249–1274 (2009)Google Scholar
  22. 22.
    Mosak, R.D.: Banach Algebras. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1975)Google Scholar
  23. 23.
    Nikol’skii, S.M.: Some inequalities for entire functions of finite degree and their application. Dokl. Akad. Nauk SSSR 76, 785–788 (1951)MathSciNetGoogle Scholar
  24. 24.
    Nikol’skii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin–Heidelberg (1975)Google Scholar
  25. 25.
    Triebel, H.: General function spaces. II. Inequalities of Plancherel–Polya–Nikol’skii-type, L p-Space of analytic functions: 0<p. J. Approx. Theory 19, 154–175 (1977)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Tuan, V.K.: On the supports of functions. Numer. Funct. Anal. Optim. 20, 387–394 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tuan, V.K., Zayed, A.I.: Generalization of a theorem of Boas to a class of integral transforms. Results Math. 38, 362–376 (2000)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Tuan, V.K., Zayed, A.I.: Paley–Wiener-type theorems for a class of integral transforms. J. Math. Anal. Appl. 266, 200–226 (2002)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tuan, V.K.: Spectrum of signals. J. Fourier Anal. Appl. 7, 319–323 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnamese Academy of Science and TechnologyCau GiayVietnam
  2. 2.Department of Mathematics, College of ScienceVietnam National UniversityThanh XuanVietnam

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