Document Type: Research Paper

Authors

School of Mathematics and computer Science, Damghan University, Damghan, Iran.

Abstract

In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective operator is replaced by unitary and invertible operators, respectively. Moreover, we show that new multi-frames (resp. multi-Riesz bases) can be obtained as linear combinations of known ones using coefficients which are operators in a certain class.

Keywords

Main Subjects

###### ##### References

[1] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Continuous frames in Hilbert space, Ann. Physics., 222 (1993), pp. 1-37.

[2] B.K. Alpert, A class of bases in $L^2$ for the sparse representation of integral operators, SIAM J. Math. Anal., 24 (1993), pp. 246-262.

[3] L. Arambasic, On frames for countably generated Hilbert $C^*$-modules, Proc. Amer. Math. Soc., 135 (2007), pp. 469-478.

[4] D. Bakic and B. Guljas, Hilbert $C^*$-modules over $C^*$-algebras of compact operators, Acta Sci. Math. (Szeged), 68 (2002), pp. 249-269.

[5] P. Balazs, M. D"orfler, N. Holighaus, F. Jaillet, and G. Velasco, Theory, implementation and applications of nonstationary Gabor frames, J. Comput. Appl. Math., 236 (2011), pp. 1481-1496.

[6] P. Balazs, B. Laback, G. Eckel, and W.A. Deutsch, Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, IEEE Trans. Audio. Speech. Language Process., 18 (2010), pp. 34-49.

[7] J.J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 5 (1998), pp. 389-427.

[8] H. B"olcskei, F. Hlawatsch, and H.G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Processing., 46 (1998), pp. 3256-3268.

[9] P. Casazza and G. Kutyniok, Finite Frames: Theory And Applications, Springer Science & Business Media, Birkhauser, 2012.

[10] P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 254 (2008), pp. 114-132.

[11] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, 2016.

[12] O. Christensen and D. Stoeva, $p$-frames in separable Banach spaces, Adv. Comput. Math., 18 (2003), pp. 117-126.

[13] N. Cotfas and J.P. Gazeau, Finite tight frames and some applications, J. Phys. A., 43 (2010), p. 193001.

[14] S. Dahlke, M. Fornasier, and T. Raasch, Adaptive Frame Methods for Elliptic Operator Equations, Adv. Comput. Math., 27 (2007), pp. 27-63.

[15] X. Dai and D.R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Amer. Math. Soc., 640, 1998.

[16] I. Daubechies, Ten lectures on wavelet, SIAM, Philadelphia, 27, 1992.

[17] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), pp. 1271-1283.

[18] M. Dorfler and H. Feichtinger, Quilted Gabor families I: Reduced multi-Gabor frames, Appl. Comput. Harmon. Anal., 356 (2004), pp. 2001-2023.

[19] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[20] M. Frank and D. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory., 48 (2002), pp. 273-314.

[21] D. Gabor, Theory of communication. Part 1: The analysis of information, Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering 93 (1946), pp. 429-441.

[22] X. Guo, Multi-frame vectors for unitary systems, Indian J. Pure Appl. Math., 43 (2012), pp. 391-409.

[23] D. Han, Tight frame approximation for multi-frames and supper-frames, J. Approx. Theory., 129 (2004), pp. 78-93.

[24] C. Heil, A Basis Theory Primer, expanded edition. Springer Science & Business Media, 2010.

[25] L. Herve, Multi-resolution analysis of multiplicity d: applications to dyadic interpolation, Appl. Comput. Harmon. Anal., 1 (1994), pp. 299-315.

[26] W. Jing, Frames in Hilbert $C^*$-modules, Ph.D. Thesis, University of Central Florida, 2006.

[27] E.C. Lance, Unitary operators on Hilbert $C^*$-modules, Bull. Lond. Math. Soc., 26 (1994), pp. 363-366.

[28] E.C. Lance, Hilbert $C^*$modules: A Toolkit for Operator Algebraists, Cambridge University Press, Cambridge, 1995.

[29] S.Q. Liu, H.L. Jin, X.O. Tang, H.Q. Lu and S.D. Ma, Boosting multi-Gabor subspaces for face recognition, In: Asian Conference on Computer Vision. Springer, Berlin, Heidelberg, 2006, 539-548.

[30] P. Majdak, P. Balazs, W. Kreuzer, and M. Dorfler, A time-frequency method for increasing the signal-to-noise ratio insystem identification with exponential sweeps, In: Proceedings of the 36th International Conference on Acoustics, Speech and Signal Processing, ICASSP 2011, Prag, 2011, 3812-3815.

[31] V.M. Manuilov and E.V. Troitsky, Hilbert $C^*$modules, Amer. Math. Soc., 2005.

[32] R. Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal., 41 (2003), pp. 1074-1100.