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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 Bang
  • Vu Nhat Huy
Article
  • 64 Downloads

Abstract

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.

Keywords

Lp - spaces Fourier transform Generalized functions 

Mathematics Subject Classification (2010)

Primary 26D10 Secondary 46E30 

Notes

Acknowledgements

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.

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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

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