- Research
- Open access
- Published:
Some equalities and inequalities for probabilistic frames
Journal of Inequalities and Applications volume 2016, Article number: 245 (2016)
Abstract
Probabilistic frames have some properties which are similar to those of frames in Hilbert space. Some equalities and inequalities have been established for traditional frames. In this paper, we give some equalities and inequalities for probabilistic frames. Our results generalize and improve the remarkable results which have been obtained.
1 Introduction
Frames are redundant systems of vectors for a Hilbert space, which can yield many different and stable representations for a given vector [1]. The frame was first introduced by Duffin and Schaeffer in the context of nonharmonic Fourier series [2]. To date, frame theory has broad applications in pure mathematics, for instance, the Kadison-Singer problem [3] and statistics [4], as well as in applied mathematics, computer science, and emerging applications.
Due to the redundancy of frames, the frame has become an essential tool in signal processing such as wireless communication [5, 6], image processing [7], coding theory [8], and sampling theory [9]. These applications led to resilience to additive noise and quantization [10, 11], resilience to erasures [12–15], and numerical stability of reconstructions [16, 17].
By viewing the frame vectors as discrete mass distributions on \(\mathbb {R} ^{N}\), being the generation of frames, probabilistic frames were developed by Ehler [18] and further expanded in [19]. Due to the connections between probability measures and frame theory, probabilistic frames are tightly related to various notions that appeared in areas such as the theory of t-designs [20], positive operator valued measures encountered in quantum computing [21, 22], and isometric measures used in the study of convex bodies [23]. Now, some excellent results of class frames have been obtained and applied successfully. It is necessary to extend some important results of conventional frames to the probabilistic frames.
In this paper, we mainly research the equalities and inequalities of probabilistic frames. Balan et al. obtained an identity when studying the optimal decomposition of Parseval frames [24], and they discovered a surprising identity for Parseval frames when working on reconstructing signal without noisy phase or its estimation in [25]. Subsequently, some authors found and improved some equalities or inequalities of the traditional frames based on the work of Balan et al.
First we will recall the definition and some properties of probabilistic frames in Hilbert spaces.
Throughout this paper \({\mathcal {H} }\) will always denote a Hilbert space, I denotes a countable indexing set and \(I_{{\mathcal {H} }}\) denotes the identity operator on \({\mathcal {H} }\). A system \(\{f_{i}\}_{i\in I}\) is called a frame for \({\mathcal {H} }\) if there exist the constants \(0 < A\le B < \infty\) such that
for all \(f\in {\mathcal {H} }\). The constants A and B are called lower and upper frame bounds, respectively. If \(A=B\), then the frame is called an A-tight frame, and if \(A=B=1\), then it is called a Parseval frame. A Bessel sequence \(\{f_{i}\}_{i\in I}\) is only required to fulfill the upper frame bound estimate but not necessarily the lower estimate.
For more details on conventional frames we refer to [1, 26].
Let I be a nonempty subset of \(\mathbb {R} ^{N}\) and let \({\mathcal {M} }({\mathcal {B} },I)\) denote the collection of probability measures on I with respect to the Borel σ-algebra \({\mathcal {B} }\).
Definition 1
A probability measure \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) is called a probabilistic frame for \({\mathcal {H} }\) if there are constants \(0 < A\le B < \infty\) such that
The constants A and B are called lower and upper probabilistic frame bounds, respectively. If \(A=B\), then the frame is called a probabilistic A-tight frame for \({\mathcal {H} }\), and if \(A=B=1\), then it is called a probabilistic Parseval frame. If only the upper inequality holds, then we call μ a Bessel measure.
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic frame. The probabilistic analysis operator is given by
The adjoint operator \(T^{*}\) of T is called the probabilistic synthesis operator which is given by
The probabilistic frame operator of μ is \(S=T^{*}T\), and one easily verifies that
is positive, self-adjoint, and invertible.
For any \(J\subset I\), we define a bounded linear operator \(S_{J}\) as
and denote \(J^{c}=I\setminus J\).
Moreover, for \(\tilde{\mu}=\mu\circ S\), we have
Using the fact that \(S^{-1}S=SS^{-1}=I_{{\mathcal {H} }}\), the reconstruction formula is given by
for all \(x\in {\mathcal {H} }\).
Definition 2
If \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) is a probabilistic frame, then \(\tilde{\mu}=\mu \circ S\) is called the probabilistic canonical dual frame of μ.
Proposition 1
If \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) is a probabilistic frame, then \(\tilde{\mu}=\mu \circ S^{1/2}\) is a probabilistic Parseval frame for \({\mathcal {H} }\).
We refer to [18, 19, 27] for more details on probabilistic frames.
In order to compare with our result, we list some important equalities as follows.
Theorem 1
[28] Let \(\{f_{i}\}_{i\in I}\) be a frame for \({\mathcal {H} }\) with canonical dual frame \(\{g_{i}\}_{g_{i}}\). Then for all \(J\subset I\) and all \(f\in {\mathcal {H} }\) we have
In the situation of Parseval frames, the authors of [28] gave the new identity which is given by
Then the general result for (1) was established in [29] as follows.
Theorem 2
Let \(\{f_{i}\}_{i\in I}\) be a frame for \({\mathcal {H} }\) with canonical dual frame \(\{g_{i}\}_{g_{i}}\). Then for all \(J\subset I\) and all \(f\in {\mathcal {H} }\), we have
Note that the above result involves the real parts of some complex number. Zhu and Wu [30] generalized the above equality to a more general form which does not involve the real parts of the complex numbers.
Theorem 3
Let \(\{f_{i}\}_{i\in I}\) be a frame for \({\mathcal {H} }\) with canonical dual frame \(\{g_{i}\}_{g_{i}}\). Then for all \(J\subset I\) and all \(f\in {\mathcal {H} }\), we have
Next, we extend these equalities to probabilistic frames.
2 The main result for probabilistic Parseval frames
In this section, we continue the work [28, 29] about probabilistic Parseval frames and obtain some important equalities and inequalities of these frames.
Lemma 1
[30]
Let P and Q be two linear bounded operators on \({\mathcal {H} }\) such that \(P+Q=I_{{\mathcal {H} }}\). Then
Then we have the following result.
Theorem 4
If \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) is a probabilistic Parseval frame for \({\mathcal {H} }\), then for all \(J\subset I\) and all \(x\in {\mathcal {H} }\), we have
Proof
Since μ is a Parseval frame, we have \(S=I_{{\mathcal {H} }}\), clearly, \(S_{J}+S_{j^{c}}=I_{{\mathcal {H} }}\). Thus, by Lemma 1, we have
 □
Note that each side of (2) is non-negative. An overlapping division of (2) is given as follows.
Proposition 2
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic Parseval frame for \({\mathcal {H} }\). For every \(J\subset I\), every \(E\subset J^{c}\), and all \(x\in {\mathcal {H} }\), we have
Proof
Applying Theorem 4 twice, then we have
 □
Corollary 1
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic Parseval frame for \({\mathcal {H} }\). For every \(J\subset I\), every \(F\subset J\), every \(E\subset J^{c}\) and all \(x\in {\mathcal {H} }\), we have
The proof of Corollary 1 is immediate.
By Proposition 1, each probabilistic A-tight frame can be turned into a probabilistic Parseval frame.
Corollary 2
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic tight Parseval frame with bound A for \({\mathcal {H} }\). For every \(J\subset I\) and every \(x\in {\mathcal {H} }\), we have
The proof of Corollary 2 is straightforward.
Next, we give a discussion of Theorem 4. From Theorem 4, for every \(J\subset I\) and every \(f\in {\mathcal {H} }\), we have
The above equality leads us to introduce some notation: \(\nu_{-}({\mathcal {U} };J)\) and \(\nu_{+}({\mathcal {U} };J)\). Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic Parseval frame for \({\mathcal {H} }\). For every \(J\subset I\), we define
Some propositions of these notations are given in the following results.
Theorem 5
The notations \(\nu_{-}({\mathcal {U} };J)\) and \(\nu_{+}({\mathcal {U} };J)\) have the following properties:
-
(i)
\(\frac{3}{4}\le\nu_{-}({\mathcal {U} };J)\le\nu_{+}({\mathcal {U} };J)\le1\);
-
(ii)
\(\nu_{-}({\mathcal {U} };J)=\nu_{-}({\mathcal {U} };J^{c})\) and \(\nu_{+}({\mathcal {U} };J)=\nu_{+}({\mathcal {U} };J^{c})\);
-
(iii)
\(\nu_{-}({\mathcal {U} };I)=\nu_{+}({\mathcal {U} };I)\) and \(\nu_{-}({\mathcal {U} };\emptyset)=\nu _{+}({\mathcal {U} };\emptyset)\).
Proof
(i) We first proof the first inequality. Since \(S_{J}+S_{J^{c}}=I_{{\mathcal {H} }}\), then we have
Hence,
with equality if and only if \(S_{J}^{2}=\frac{1}{2}I_{{\mathcal {H} }}\). Therefore, for every \(x\in {\mathcal {H} }\) and \(x\ne0\), we have
with equality if and only if \(S_{J}^{2}=\frac{1}{2}I_{{\mathcal {H} }}\).
Next, we prove the second inequality. Since
we have
(ii) and (iii) follow from Theorem 4. □
Corollary 3
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic Parseval frame for \({\mathcal {H} }\). For every \(J\subset I\) and every \(x\in {\mathcal {H} }\), \(\nu_{-}({\mathcal {U} };J)=\nu_{+}({\mathcal {U} };J)=1\) if and only if \(S_{J}x=S_{J}^{2}x\).
Proof
From the definition of ν, \(\nu_{-}({\mathcal {U} };J)=\nu _{+}({\mathcal {U} };J)=1\) if and only if
And
which proves the results. □
3 The main result for probabilistic frames
In this section, we extend some results of conventional frames to general probabilistic frames.
Theorem 6
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic frame for \({\mathcal {H} }\) with probabilistic canonical dual frame μ̃. For every \(J\subset I\) and every \(x\in {\mathcal {H} }\), we have
Proof
The equality in Theorem 6 can be written as
Also
Simultaneously,
Since \(S=S_{J}+S_{J^{c}}\), it follows that \(I_{{\mathcal {H} }}=S^{-1}S_{J}+S^{-1}S_{J^{c}}\). From the proof of Theorem 5, we have
Moreover, for every \(x,z\in {\mathcal {H} }\), we have
If we choose z to be \(z=Sx\), by the equalities (3) and (4), (5) is equal to
Hence, the proof is completed. □
In the case of general probabilistic frames, we define notations as follows:
These notations of general probabilistic frames also satisfy the properties (i)-(iii) in Theorem 5. We give a detailed proof for the property (i).
Proposition 3
The notations \(\nu_{-}^{\prime}({\mathcal {U} };J)\) and \(\nu_{+}^{\prime}({\mathcal {U} };J)\) satisfy
Before the proof of Proposition 3, we need the following lemma.
Lemma 2
[29]
If \(P,Q\) are self-adjoint operators on \({\mathcal {H} }\) satisfying \(P+Q=I_{{\mathcal {H} }}\), then
for all \(x\in {\mathcal {H} }\).
Proof of Proposition 3
First, we prove the left inequality. Since \(S=S_{J}+S_{J^{c}}\), it follows that
By Lemma 2, we get
Replacing x by \(S^{1/2}x\) for the above equality, then we have
Since
we have \(\nu_{+}^{\prime}({\mathcal {U} };J)\ge\nu_{-}^{\prime}({\mathcal {U} };J)\ge\frac{3}{4}\).
The right inequality is also true. In fact,
It follows \(\nu_{+}^{\prime}({\mathcal {U} };J)\le1\). The proof is completed. □
Next, we give a generalization of the equality from Theorem 4 for general probabilistic frames with probabilistic canonical dual frames.
Theorem 7
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic frame for \({\mathcal {H} }\) with the probabilistic canonical dual frame μ̃. Let \(z=Sy\), for every \(J\subset I\) and every \(x\in {\mathcal {H} }\), we have
Proof
Let \(z=Sy\), for every \(x\in {\mathcal {H} }\), we have
For every \(J\subset I\), we define the operator \(V_{J}\) as follows:
It follows that \(V_{J}+V_{J^{c}}=I_{{\mathcal {H} }}\) by (6). Thus, by Lemma 1, we have
 □
If we take the real part on both sides of equality in Theorem 7, we can get a more general result.
Theorem 8
Let \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) be a probabilistic frame for \({\mathcal {H} }\) with probabilistic canonical dual frame μ̃. Let \(z=Sy\), for every \(J\subset I\), every continue bounded sequence \(\lbrace b_{i} \rbrace n L^{2}(I)\) and every \(x\in {\mathcal {H} }\), we have
The proof of Theorem 8 is immediate.
For example, we can take \(b_{i}=1\) if \(i\in J\) and \(b_{i}=0\) if \(i\in J^{c}\). As a special case we have the following result.
Corollary 4
If \(\mu\in {\mathcal {M} }({\mathcal {B} },I)\) is a probabilistic A-tight frame for \({\mathcal {H} }\) with probabilistic canonical dual frame μ̃. Let \(z=Sy\), for every \(J\subset I\), every continue bounded sequence \(\lbrace b_{i} \rbrace n L^{2}(I)\) and every \(x\in {\mathcal {H} }\), we have
4 Conclusions
In this paper, we mainly study some equalities and inequalities for probabilistic frames. We extend some good results of frames to probabilistic frames, and we obtain some new results because not all of properties of probabilistic frames are similar to those of traditional frames. Our results generalize and improve the remarkable results which have been established.
References
Christensen, O: An Introduction to Frames and Riesz Bases. Birkhäuser, Basel (2003)
Duffin, RJ, Schaeffer, AC: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72(2), 341-366 (1952)
Casazza, PG, Fickus, M, Tremain, JC, Weber, E: The Kadison-Singer problem in mathematics and engineering: a detailed account. Contemp. Math. 414, 299 (2006)
Ehler, M, Galanis, J: Frame theory in directional statistics. Stat. Probab. Lett. 81(8), 1046-1051 (2011)
Heath, RK Jr., Bölcskei, H, Paulraj, AJ: Space-time signaling and frame theory. In: 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2001. Proceedings (ICASSP’01), vol. 4, pp. 2445-2448. IEEE, New York (2001)
Strohmer, T: Approximation of dual Gabor frames, window decay, and wireless communications. Appl. Comput. Harmon. Anal. 11(2), 243-262 (2001)
Kutyniok, G, Labate, D: Shearlets: Multiscale Analysis for Multivariate Data. Birkhäuser, Basel (2012)
Strohmer, T, Heath, RW: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257-275 (2003)
Eldar, YC: Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors. J. Fourier Anal. Appl. 9(1), 77-96 (2003)
Daubechies, I: Ten Lecture on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992)
Goyal, VK, Vetterli, M, Thao, NT: Quantized overcomplete expansions in IR N: analysis, synthesis, and algorithms. IEEE Trans. Inf. Theory 44(1), 16-31 (1998)
Han, D, Sun, W: Reconstruction of signals from frame coefficients with erasures at unknown locations. IEEE Trans. Inf. Theory 60(7), 4013-4025 (2014)
Leng, J, Han, D: Optimal dual frames for erasures II. Linear Algebra Appl. 435(6), 1464-1472 (2011)
Leng, J, Han, D, Huang, T: Optimal dual frames for communication coding with probabilistic erasures. IEEE Trans. Signal Process. 59(11), 5380-5389 (2011)
Leng, J, Han, D, Huang, T: Probability modelled optimal frames for erasures. Linear Algebra Appl. 438(11), 4222-4236 (2013)
Benedetto, JJ: Irregular sampling and frames. Wavelets: A Tutorial in Theory and Applications 2, 445-507 (1992)
Feichtinger, HG, Gröchenig, K: Theory and practice of irregular sampling. Wavelets: mathematics and applications 1994, 305-363 (1994)
Ehler, M: Random tight frames. J. Fourier Anal. Appl. 18(1), 1-20 (2012)
Ehler, M, Okoudjou, KA: Minimization of the probabilistic p-frame potential. J. Stat. Plan. Inference 142(3), 645-659 (2012)
Delsarte, P, Goethals, J-M, Seidel, JJ: Spherical codes and designs. Geom. Dedic. 6(3), 363-388 (1977)
Albini, P, De Vito, E, Toigo, A: Quantum homodyne tomography as an informationally complete positive-operator-valued measure. J. Phys. A, Math. Theor. 42(29), 295302 (2009)
Davies, EB: Quantum Theory of Open Systems. Academic Press, New York (1976)
Giannopoulos, A, Milman, V: Extremal problems and isotropic positions of convex bodies. Isr. J. Math. 117(1), 29-60 (2000)
Balan, R, Casazza, PG, Kutyniok, G: Decompositions of frames and a new frame identity. Proc Spie. 379-388 (2005)
Balan, R, Casazza, P, Edidin, D: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345-356 (2006)
Casazza, PG: The art of frame theory (1999). arXiv:math/9910168
Christensen, O, Stoeva, DT: P-frames in separable Banach spaces. Adv. Comput. Math. 18(2), 117-126 (2003)
Balan, R, Kutyniok, G: A new identity for Parseval frames. Proc. Am. Math. Soc. 135(135), 1007-1015 (2007)
Găvruţa, P: On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl. 1(321), 469-478 (2006)
Zhu, X, Wu, G: A note on some equalities for frames in Hilbert spaces. Appl. Math. Lett. 23(7), 788-790 (2010)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11271001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This work was carried out in collaboration among the authors. All authors made a good contribution to design the research. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, D., Leng, J., Huang, T. et al. Some equalities and inequalities for probabilistic frames. J Inequal Appl 2016, 245 (2016). https://doi.org/10.1186/s13660-016-1183-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-016-1183-0