Localization Operators on Sobolev Spaces

Document Type : Research Paper


Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran.



In this paper,  we discuss some generalizations coming from wavelet transform on Sobolev spaces. In particular, we introduce the bounded localization operators on Sobolev spaces which are related to multi-dimensional wavelet transform on Sobolev spaces. Moreover, we propose the localization operators on Sobolev spaces are in $p$-Schatten class and they are compact. Finally, we give the boundedness and compactness of localization operators on Sobolev spaces with two admissible wavelets.


[1] S.T. Ali, J-P. Antoine and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 
[2] J-P. Aubin, Applied Functional Analysis, New York, John Wiley, CRC Press, 2000.
[3] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Transactions on 
      Information Theory, 36(5) (1990), 961-1005.
[4] I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory,
      34, (1988), 605-612.
[5] F. De Mari, H.G. Feichtinger and K. Nowak, Uniform eigenvalue estimates for time-frequency localization operators, J.
      London. Math. Soc., 7(2) (2021), 211-218.
[6] G.B. Folland, Real Analysis: Modern Techniques and Their Applications, New York, John Wiley, 1999.
[7] F. Esmaeelzadeh, Multi-dimensional wavelets on Sobolev spaces, Khayyam Journal of Mathematics, 65(3) (2021), 
[8] W. Rudin, Functional Analysis, Tata McGraw-Hill, New York, 1973.
[9] M.W. Wong, Wavelet Transform and Localization Operator, Verlag, Basel- Boston- Berlin, 2002.