Matrix representation of operators using frames

Sampling Theory in Signal and Image Processing

Volumn 7, Issue 1, 2008, Pages 39-54

  Balazs, Peter ^a  

^a ERICH SCHMID INSTITUTE OF MATERIALS SCIENCE   (Austria)

Author keywords

Discrete expansion; Frames; Frobenius matrices; Hilbert Schmidt operators; Matrix representation; Operators; Riesz bases

Indexed keywords

EID: 39449098506     PISSN: 15306429     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (30)

1. A. Aldroubi and K. Gröchenig, Non-uniform sampling and reconstruction in shift-invariant spaces. SIAM Review, 43, 585-620, 2001.
2. S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalization, Graduate Texts in Contemporary Physics, Springer, New York, 2000.
3. P. Balazs, Regular and Irregular Gabor Multipliers with Application to Psychoacoustic Masking. Ph.D. thesis, University of Vienna, June 2005.
4. P. Balazs, Basic definition and properties of Bessel multipliers. Journal of Mathematical Analysis and Applications, 325(1), 571-585, January 2007.
5. P. Balazs, W. Kreuzer, and H. Waubke, A stochastic 2d-model for calculating vibrations in liquids and soils, Journal of Computational Acoustics, to appear.
6. P. Balazs, Hilbert-Schmidt operators and frames - classification, approximation by multipliers and algorithms, International Journal of Wavelets, Multiresolution and Information Processing, to appear.
7. J. Benedetto and P. Ferreira, Modern Sampling Theory: Mathematics and Applications, Birkhäuser, 2001.
8. H. Bölcskei, F. Hlawatsch, and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Processing, 46(12), 3256-3268, 1998.
9. P. G. Casazza, The art of frame theory, Taiwanese J. Math., 4(2), 129-202, 2000.
10. O. Christensen, Frames and pseudo-inverses, J. Math. Anal. Appl, 195(2), 401-414, 1995.
11. O. Christensen, An Introduction To Frames And Riesz Bases, Birkhäuser, 2003.
12. O. Christensen and T. Strohmer, The finite section method and problems in frame theory, Journal of Approximation Theory, 133(2), 221-237, 2005.
13. J. B. Conway, A Course in Functional Analysis. Graduate Texts in Mathematics, Springer, New York, 2nd edition, 1990.
14. I. Daubechies, Ten Lectures On Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM Philadelphia, 1992.
15. R. J. Duffin and A. C. Schaeffer, A class of nonhannonic Fourier series, Trans. Amer. Math. Soc., 72, 341-366, 1952.
16. H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Chapter 7 of [17], 233-266, 1998.
17. H. G. Feichtinger and T. Strohmer, Gabor Analysis and Algorithms - Theory and Applications, Birkhäuser, Boston, 1998.
18. M. Fornasier and K. Gröchenig, Intrinsic localization of frames, Constructive Approximation, 22(3), 395-415, 2005.
19. I. Gohberg, S. Goldberg, and M. Kaashoek, Basic Classes of Linear Operators, Birkhäuser, 2003.
20. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
21. K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam., 22, 2006.
22. R. Kronland-Martinet and A. Grossmann, Application of time-frequency and timescale methods (wavelet transform) to the analysis, synthesis and transformation of natural sounds. In G. De Poli, A. Piccialli, and C. Roads, editors, Representations of musical signals, 45-85. MIT Press, Cambridge, 1991.
23. G. Matz and F. Hlawatsch, Linear Time-Frequency Filters: On-line Algorithms and Applications, Chapter 6 of A. Papandreou-Suppappola, ed., Application in Time-Frequency Signal Processing, 205-271, CRC Press, Boca Raton, FL, 2002.
24. G. Pfander and D. Walnut, Operator identification and Feichtinger's' algebra, Sampl. Theory Signal Image Process., 5(2), 183-200, May 2006.
25. R. Schatten, Norm Ideals of Completely Continious Operators, Springer, Berlin, 1960.
26. T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comp. Harm. Anal., 20 (2), 237-249, March 2006.
27. G. Teschke, Multi-frame representations in linear inverse problems with mixed multi-constraints, Appl. Comp. Harm. Anal., 22 (1), 43-60, Jan. 2006.
28. L. N. Trefethen and D. Ban III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
29. M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, 1995.
30. D. Werner, Funktionalanalysis, Springer, Berlin, 1995.