Spectral Properties for Pseudodifferential Operators via Weighted Modulation Spaces

A. Askari-Hemmat, Z. Rahbani


In this paper we deal with this question: considering spectral representation of a positive trace-class integral operator, if its orthonormal eigenvectors are in modulation space $M^p_m$? This question actually provide a new framework for studying the connection between operator theory and modulation spaces. Here we use some Schatten class properties of seudodifferential operators to give a positive answer to this question. Also we investigate convergence conditions for eigenvectors of such operators.


Frames; Gabor systems; Modulation spaces; Wilson bases; Spectral representation; trace-class operators

Full Text:



P. Boggiatto and J. Toft, Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces, Appl. Anal. 84(3) (2005), 269–282.

P. Boggiatto and J. Toft, Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces, Adv. Math., 217(1) (2008), 305–333.

R.R. Coifman andY. Meyer, Au dela des Opérateurs Pseudo-différentiels, Asterisque 57, 1978.

E. Cordero. Gelfand-Shilov windows for weighted modulation spaces. Integral Transforms Spec. Funct., 2006.

I. Daubechies, S. Jaffard and J. L. Journé, A simple Wilson orthonormal basis with exponential decay, SIAM J.Math. Anal. 22 (1991), 554–573.

O. El Fallah, N. K. Nikolskij and M. Zarrabi, Resolvent estimates in Beurling-Sobolev algebras, St. Petersbg. Math. J. 10(6) (1998), 901–964.

H.G. Feichtinger, English translation of: Gewichtsfunktionen auf lokalkompakten Gruppen, Sitzungsber.d.österr. Akad.Wiss. 188 (1979), 451–471.

H.G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), 269–289.

H.G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical report, January 1983.

H.G. Feichtinger, Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process. 5(2) (2006), 109–140.

H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307–340.

H.G. Feichtinger and K. Gröchenig, Gabor wavelets and the Heisenberg group: Gabor Expansions and Short Time Fourier transform from the group theoretical point of view, In C. Chui, editor, Wavelets –A Tutorial in Theory and Applications, Volume 2 of Wavelet Anal. Appl., pp. 359–397, Academic Press, Boston, 1992.

H.G. Feichtinger,K. GröchenigandD.F. Walnut, Wilson basesand modulation spaces, Math. Nachr. 155 (1992), 7–17.

H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, In H. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms. Theory and Applications., Applied and Numerical Harmonic Analysis, pp. 233–266; pp. 452–488, Boston, MA, 1998, Birkhäuser Boston.

H.G. Feichtinger andT. Strohmer, Advances in Gabor Analysis, Birkhäuser, Basel, 2003.

K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. Birkhäuser Boston, Boston, MA, 2001.

K. Gröchenig, Weight functions in time-frequency analysis, In L. Rodino and et al., editors, Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis,Volume 52, pp. 343–366, 2007.

C. Heil, Integral operators, pseudodifferential operators, and Gabor frames, In H. Feichtinger and T. Strohmer (editors), Advances in Gabor analysis, Applied and Numerical Harmonic Analysis, pp. 153–169, Birkhäuser, Basel, 2003.

C. Heil and D. Larson, Operator Theory and Modulation Spaces, Frames and Operator Theory in Analysis and Signal Processing, pp. 137–150, 2008.

R. Kerman and E. Sawyer, Convolution algebras with weighted rearrangement-invariant norm, Studia Math. 108(2) (1994), 102–126.

D. Labate, J. Math. Anal. Appl., 262(1) (2001), 242–255.

F. Luef and Z. Rahbani, Pseudodifferential operators with symbols in generalized shubin classes and an application to Landau-Weyl operators. Banach J. Math. Anal. 5(2) (2011), 59–72.

N. Nikolski, Spectral synthesis for the shift operator, and zeros in certain classes of analytic functions that are smooth up to the boundary, Dokl. Akad. Nauk SSSR 190 (1970), 780–783.

S. Pilipovic and N. Teofanov, Wilson bases and ultramodulation spaces, Math. Nachr. 242 (2002), 179–196.

DOI: http://dx.doi.org/10.26713%2Fcma.v3i3.210


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905