Vietnam Journal of Mathematics

, Volume 45, Issue 4, pp 737–746 | Cite as

Local Spectral Formula for Integral Operators on \(L_{p}({\mathbb T})\)

  • Ha Huy BangEmail author
  • Vu Nhat Huy


Let \(1\leq p\leq \infty \), \(f\in L_{p}({\mathbb T})\) and \(0 \notin \text {supp} \hat {f}\). Then, in this paper, we obtain the following local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\), the space of 2π-periodic functions belonging to L p (−π,π): \( \lim _{n\rightarrow \infty } \|I^{n} f\|_{p,{\mathbb T}}^{1/n}= \sigma ^{-1}, \) where \(\sigma =\min \{ |k| : k \in \text {supp} \hat {f} \}, If(x)={{\int }_{0}^{x}} f(t) dt -c_{f}, x\in \mathbb {R}\) and the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0\). The local spectral formula for polynomial integral operators on \(L_{p}({\mathbb T})\) is also given.


Lp - spaces Fourier transform Generalized functions 

Mathematics Subject Classification (2010)

Primary 26D10 Secondary 46E30 



The research of V. N. Huy is funded by the Vietnam National University, Hanoi (VNU) under project number QG.16.08. A part of this work was done when V. N. Huy is working at the Vietnam Institute for Advanced Study in Mathematics (VIASM); the author would like to thank the VIASM for providing a fruitful research environment and working condition. The authors would like to thank the referees for useful remarks and comments.


  1. 1.
    Abreu, L.D.: Real Paley-Wiener theorems for the Koornwinder-Swarttouw q-Hankel transform. J. Math. Anal. Appl. 334, 223–231 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersen, N.B.: On real Paley–Wiener theorems for certain integral transforms. J. Math. Anal. Appl. 288, 124–135 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andersen, N.B.: Real Paley–Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. Pac. J. Math. 213, 1–13 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andersen, N.B.: Real Paley–Wiener theorems for the Hankel transform. J. Fourier Anal. Appl. 12, 17–25 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andersen, N.B., De Jeu, M.: Real Paley-Wiener theorems and local spectral radius formulas. Trans. Am. Math. Soc. 362, 3613–3640 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bang, H.H.: A property of infinitely differentiable functions. Proc. Am. Math. Soc. 108, 73–76 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bang, H.H.: Functions with bounded spectrum. Trans. Am. Math. Soc. 347, 1067–1080 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bang, H.H.: An inequality of Bohr and Favard for Orlicz spaces. Bull. Pol. Acad. Sci. Math. 49, 381–387 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bang, H.H., Huy, V.N.: Behavior of the sequence of norm of primitives of a function. J. Approx. Theory 162, 1178–1186 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)zbMATHGoogle Scholar
  11. 11.
    Chettaoui, C., Trimèche, K.: New type Paley–Wiener theorems for the Dunkl transform on R. Integral Trans. Spec. Funct. 14, 97–115 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garth Dales, H., Aeina, P., Eschmeier, J., Laursen, K., Willis, G.: Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press, Cambridge (2003)Google Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    De Jeu, M.: Paley–Wiener theorems for the Dunkl transform. Trans. Am. Math. Soc. 358, 4225–4250 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tuan, V.K., Zayed, A.: Paley–Wiener-type theorems for a class of integral transforms. J. Math. Anal. Appl. 266(1), 200–226 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tuan, V.K.: Spectrum of signals. J. Fourier Anal. Appl. 7, 319–323 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vladimirov, V.S.: Methods of the Theory of Generalized Functions. Taylor & Francis, London, New York (2002)zbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

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

Personalised recommendations