Pseudo-Differential Operators on ℤ
A necessary and sufficient condition is imposed on the symbols σ: ℤ×S 1→ℂ to guarantee that the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) are Hilbert-Schmidt. A special sufficient condition on the symbols σ: ℤ×S 1→ℂ for the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) to be bounded is given. Sufficient conditions are given on the symbols σ: ℤ×S 1→ℂ to ensure the boundedness and compactness of the corresponding pseudo-differential operators T σ: L p (ℤ)→L p (ℤ) for 1≨p<∞. Norm estimates for the pseudo-differential operators T σ are given in terms of the symbols σ. The almost diagonalization of the pseudo-differential operators is then shown to follow from the sufficient condition for the L p -boundedness.
KeywordsPseudo-differential operators Hilbert-Schmidt operators L2-boundedness Lp-boundedness Lp-compactness almost diagonalization
Mathematics Subject Classification (2000)Primary 47G30 Secondary 65R10 65T50
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- E. Cordero, F. Nicola and L. Rodino, Fourier integral operators and Gabor frames, Trans. Amer. Math. Soc., to appear.Google Scholar
- S. Molahajloo and M. W. Wong, Pseudo-differential operators on S 1, in New Developments in Pseudo-Differential Operators, Editors: L. Rodino and M. W. Wong, Birkhäuser, 2008, 297–306.Google Scholar
- M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M. W. Wong and H. Zhu, Birkhäuser, 2007, 87–105.Google Scholar
- J. Saranen and G. Vainikko, Periodic Integral and Pseudo-Differential Equations with Numerical Approximation, Springer-Verlag, 2002.Google Scholar
- M. W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999.Google Scholar
- M. W. Wong and H. Zhu, Matrix representations and numerical computations of wavelet multipliers, in Wavelets, Multiscale Systems and Hypercomplex Analysis, Editor: D. Alpay, Birkhäuser, 2006, 173–182.Google Scholar
- A. Zygmund, Trigonometric Series, Third Edition, Volumes I & II Combined, Cambridge University Press, 2002.Google Scholar