A note on finite dual frame pairs. (English) Zbl 1365.42023

Let \(\mathbb{K}^d\) denote either one of the finite dimensional spaces \(\mathbb{R}^d\) or \(\mathbb{C}^d\). A sequence \(\{e_n\}_{n=1}^N\subset \mathbb{K}^d\) is said to be a frame for \(\mathbb{K}^d\), with frame bounds \(A,B>0\), provided that \[ A\|x\|^2\leq \sum_{n=1}^N|\langle x,e_n\rangle|^2\leq B\|x\|^2,\qquad x\in \mathbb{K}^d. \] The frame \(\{e_n\}_{n=1}^N\) is said to be tight if one can take \(A=B\) in the above inequality.
For any given frame \(\{e_n\}_{n=1}^N\), it is always possible to find at least one sequence \(\{f_n\}_{n=1}^N\subset \mathbb{K}^d\) such that \[ x=\sum_{n=1}^N\langle x,e_n\rangle f_n=\sum_{n=1}^N\langle x,f_n\rangle e_n,\qquad x\in \mathbb{K}^d. \] This \(\{f_n\}_{n=1}^N\) is also a frame and the sequences \(\{e_n\}_{n=1}^N\) and \(\{f_n\}_{n=1}^N\) are called dual frames. A tight frame \(\{e_n\}_{n=1}^N\) has \(\{\frac{1}{A}e_n\}_{n=1}^N\) for (its canonical) dual frame. The following important result characterizes the possible norms of elements of a tight frame:
Given a sequence \(\{a_n\}_{n=1}^N\subset [0,\infty)\) with \(N\geq d\), (1) there exists a tight frame \(\{e_n\}_{n=1}^N\) for \(\mathbb{K}^d\) such that \(a_n=\langle e_n,e_n\rangle\), \(n=1,\ldots,N\), if and only if (2) \(\max_{1\leq n\leq N}a_n\leq \frac{1}{d}\sum_{n=1}^Na_n\).
Inequality (2) is known as the fundamental inequality of tight frames.
The paper extends this result to a pair of dual frames \(\{e_n\}_{n=1}^N\) and \(\{f_n\}_{n=1}^N\) by characterizing the possible values of \(\langle e_n,f_n\rangle\). This is Theorem 3.1 of the paper, which states that given \(\{\alpha_n\}_{n=1}^N\subset\mathbb{K}\) with \(N>d\), the following statements are equivalent:
There exist dual frames \(\{e_n\}_{n=1}^N\) and \(\{f_n\}_{n=1}^N\) for \(\mathbb{K}^d\) such that \(\alpha_n=\langle e_n,f_n\rangle\), \(1\leq n\leq N\).
There exists a tight frame \(\{g_n\}_{n=1}^N\) and a corresponding dual frame \(\{h_n\}_{n=1}^N\) for \(\mathbb{K}^d\) such that \(\alpha_n=\langle g_n,h_n\rangle\), \(1\leq n\leq N\).
The assumption \(N>d\) is due to the fact that for \(\{e_n\}_{n=1}^N\) to be a frame for \(\mathbb{K}^d\) it is necessary that \(N\geq d\), and if \(N=d\), any frame \(\{e_n\}_{n=1}^N\) is a basis and has a unique dual frame, namely, \(\{\|e_n\|^{-2}e_n\}_{n=1}^N\), in which case \(\alpha_n=1\) for all \(n\).
The paper then studies to a good extent the frames \(\{e_n\}_{n=1}^N\) for \(\mathbb{K}^d\) having (what the authors name) the full range property, that is, those frames \(\{e_n\}_{n=1}^N\) such that for every \(\{\alpha_n\}_{n=1}^N\subset\mathbb{K}\) with \(\sum_{n=1}^N\alpha_n=d\), one can find a dual frame \(\{f_n\}_{n=1}^N\) satisfying that \(\alpha_n=\langle e_n,f_n\rangle\) for \(1\leq n\leq N\).


42C15 General harmonic expansions, frames
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[1] John J. Benedetto, Alexander M. Powell, and Özgür Yılmaz, Sigma-Delta (\Sigma \Delta ) quantization and finite frames, IEEE Trans. Inform. Theory 52 (2006), no. 5, 1990 – 2005. · Zbl 1285.94014
[2] John J. Benedetto and Matthew Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003), no. 2-4, 357 – 385. Frames. · Zbl 1028.42022
[3] James Blum, Mark Lammers, Alexander M. Powell, and Özgür Yılmaz, Sobolev duals in frame theory and sigma-delta quantization, J. Fourier Anal. Appl. 16 (2010), no. 3, 365 – 381. · Zbl 1200.42019
[4] Peter G. Casazza, Matthew Fickus, Jelena Kovačević, Manuel T. Leon, and Janet C. Tremain, A physical interpretation of tight frames, Harmonic analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, pp. 51 – 76. · Zbl 1129.42418
[5] Peter G. Casazza, Custom building finite frames, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 61 – 86. · Zbl 1082.42024
[6] P.G. Casazza and M. Leon, Frames with a given frame operator. Preprint. http://www.math.missouri.edu/\( \sim \)pete/pdf/PM2.pdf · Zbl 1225.42021
[7] Peter G. Casazza and Nicole Leonhard, Classes of finite equal norm Parseval frames, Frames and operator theory in analysis and signal processing, Contemp. Math., vol. 451, Amer. Math. Soc., Providence, RI, 2008, pp. 11 – 31. · Zbl 1210.42047
[8] Peter G. Casazza and Jelena Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003), no. 2-4, 387 – 430. Frames. · Zbl 1035.42029
[9] Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. · Zbl 1017.42022
[10] Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001), no. 3, 203 – 233. · Zbl 0992.94009
[11] Mark Lammers and Anna Maeser, An uncertainty principle for finite frames, J. Math. Anal. Appl. 373 (2011), no. 1, 242 – 247. · Zbl 1203.42040
[12] Mark Lammers, Alexander M. Powell, and Özgür Yılmaz, Alternative dual frames for digital-to-analog conversion in sigma-delta quantization, Adv. Comput. Math. 32 (2010), no. 1, 73 – 102. · Zbl 1181.94050
[13] C.S. Güntürk, M. Lammers, A.M. Powell, R. Saab and Ö. Yılmaz, Sobolev duals for random frames and \( \Sigma \Delta \) quantization of compressed sensing measurements. Preprint, 2010. · Zbl 1273.41020
[14] Shidong Li, On general frame decompositions, Numer. Funct. Anal. Optim. 16 (1995), no. 9-10, 1181 – 1191. · Zbl 0849.42023
[15] Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in \?\(_{2}\)(\?^{\?}), Duke Math. J. 89 (1997), no. 2, 237 – 282. · Zbl 0892.42017
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