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On the duality of fusion frames. (English) Zbl 1127.46016
A system $(f_i)_{i\in I}$ is a frame in a Hilbert space $\Cal H$ if there are positive constants $A$ and $B$ such that $A\Vert f\Vert ^2 \leq \sum_{i\in I} |\langle f,f_i\rangle\vert ^2 \leq B\Vert f\Vert ^2$ for all $f\in \Cal H$. A system $\Cal V= ((V_i,V_i))_{i\in I}$ is a fusion frame or a frame of subspaces if $$ A\Vert f\Vert ^2 \leq \sum_{i\in I} v_i^2 \Vert \pi_{V_i}(f)\Vert^2 \leq B\Vert f\Vert ^2 $$ where $\pi_V$ is the orthogonal projection onto the subspace $V$. One of the main results of this paper is a proof that the dual fusion frame $((S_{\Cal V}^{-1}V_i, v_i))_{i\in I}$ (with $S_{\Cal V}$ the frame operator given by $\sum_{i\in I} v_i\pi_{V_i}(f)$) is indeed a fusion frame. Other results deal with alternate duals, i.e., systems $\Cal W=((W_i,w_i))_{i\in I}$ so that $f = \sum_{i\in I} v_iw_i \pi_{W_i}S_{\Cal V}^{-1}\pi_{V_i}(f)$, and frame operators for a pair of two Bessel fusion sequences (where only the upper bound above is required to hold).

46C15Characterizations of Hilbert spaces
42C99Non-trigonometric Fourier analysis
Full Text: DOI
[1] Ali, S. T.; Antoiné, J. P.; Gazeau, J. P.: Continuous frames in Hilbert space. Ann. of phys. 222, 1-37 (1993)
[2] Asgari, M. S.; Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. math. Anal. appl. 308, No. 2, 541-553 (2005) · Zbl 1091.46006
[3] Baggett, L. W.; Larson, D. R.: The functional and harmonic analysis of wavelets and frames. Contemp. math. 247 (1999) · Zbl 0930.00049
[4] Casazza, P. G.: The art of frame theory. Taiwanese J. Math. 4, 129-201 (2000) · Zbl 0966.42022
[5] Casazza, P. G.; Kutyniok, G. K.: Frames of subspaces. Contemp. math. 345 (2004) · Zbl 1058.42019
[6] P.G. Casazza, G.K. Kutyniok, S. Li, Fusion frames and distributed processing, preprint · Zbl 1258.42029
[7] Christensen, O.: An introduction to frames and Riesz bases. Appl. numer. Harmon. anal. (2003) · Zbl 1017.42022
[8] Daubechies, I.: Ten lectures on wavelets. (1992) · Zbl 0776.42018
[9] Duffin, J.; Schaeffer, A. C.: A class of nonharmonic Fourier series. Trans. amer. Math. soc. 72, 341-366 (1952) · Zbl 0049.32401
[10] Fornasier, M.: Decompositions of Hilbert spaces: local construction of global frames. Proc. int. Conf. on constructive function theory, 275-281 (2003) · Zbl 1031.42035
[11] Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. math. Anal. appl. 289, No. 1, 180-199 (2004) · Zbl 1058.46009
[12] Găvruţa, P.: On some identities and inequalities for frames in Hilbert spaces. J. math. Anal. appl. 321, 469-478 (2006)
[13] Han, D.; Larson, D. R.: Frames, bases and group representations. Mem. amer. Math. soc. 147, No. 697 (2000) · Zbl 0971.42023
[14] Heil, C.; Walnut, D.: Continuous and discrete wavelet transforms. SIAM rev. 31, No. 4, 628-666 (1989) · Zbl 0683.42031
[15] Hernández, E.; Weiss, G.: A first course on wavelets. (1996) · Zbl 0885.42018
[16] Kaiser, G.: A friendly guide to wavelets. (1994) · Zbl 0839.42011
[17] Mallat, S.: A wavelet tour of signal processing. (1998) · Zbl 0937.94001
[18] Meyer, Y.: Ondelettes. (1990) · Zbl 0694.41037
[19] Sun, W.: G-frames and g-Riesz bases. J. math. Anal. appl. 322, 437-452 (2006) · Zbl 1129.42017