Finite Frames: Theory and Applications by Peter G. Casazza, Gitta Kutyniok

By Peter G. Casazza, Gitta Kutyniok

Hilbert house frames have lengthy served as a precious device for sign and snapshot processing as a result of their resilience to additive noise, quantization, and erasures, in addition to their skill to trap helpful sign features. extra lately, finite body idea has grown into an enormous examine subject in its personal correct, with a myriad of purposes to natural and utilized arithmetic, engineering, laptop technology, and different parts. The variety of examine guides, meetings, and workshops in this subject has elevated dramatically during the last few years, yet no survey paper or monograph has but seemed at the subject.

Edited by way of of the major specialists within the box, Finite Frames goals to fill this void within the literature through offering a accomplished, systematic research of finite body conception and functions. With rigorously chosen contributions written by means of hugely skilled researchers, it covers issues including:

* Finite body Constructions;
* optimum Erasure Resilient Frames;
* Quantization of Finite Frames;
* Finite Frames and Compressed Sensing;
* team and Gabor Frames;
* Fusion Frames.

Despite the diversity of its chapters' resource and content material, the book's notation and terminology are unified all through and supply a definitive photograph of the present country of body theory.

With a extensive variety of purposes and a transparent, complete presentation, this e-book is a hugely beneficial source for graduate scholars and researchers throughout disciplines equivalent to utilized harmonic research, electric engineering, quantum computing, medication, and extra. it's designed for use as a supplemental textbook, self-study consultant, or reference book.

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Extra resources for Finite Frames: Theory and Applications

Sample text

N . N ∗ (iii)⇒(i) Since (Aeˆi )M i=1 is a spanning set for 2 and T = U A, it follows that M ∗ N (T eˆi )i=1 forms a frame for H . Its analysis operator is given by T , since for all x ∈ H N, x, T ∗ eˆi M i=1 = T x, eˆi M i=1 = T x. Moreover, T ∗ T ej = U AA∗ U ∗ ej = U diag(λ1 , . . , λN )fj = λj Ufj = λj ej , which completes the proof. 3 Gramian Operator N with analysis operator T . The previous subsecLet (ϕi )M i=1 be a frame for H tion was concerned with properties of the frame operator defined by S = T ∗ T : H N → H N .

In particular, there exist equal norm Parseval frames with M elements in H N for every N ≤ M. Proof We define the norm of the to-be-constructed frame to be c, where c2 = 1 M N λj . 13(ii) are satisfied for ci = c for all i = 1, 2, . . , M. The definition of c immediately implies the second condition. For the first condition, we observe that c12 = c2 = 1 M N λj ≤ λ1 . j =1 Hence this condition holds for j = 1. Now, toward a contradiction, assume that there exists some k ∈ {2, . . , k−1 k−1 cj2 = (k − 1)c2 ≤ j =1 k λj , k cj2 = kc2 > but j =1 j =1 λj .

M} \ {i0 }, i0 ∈ {1, . . , M}. We can immediately make the following useful observations. 2 Let (ϕi )M i=1 be a family of vectors in H . M (i) If (ϕi )M i=1 is an orthonormal basis, then (ϕi )i=1 is a Parseval frame. The converse is not true in general. N if and only if it is a spanning set for H N . (ii) (ϕi )M i=1 is a frame for H M (iii) (ϕi )i=1 is a unit norm Parseval frame if and only if it is an orthonormal basis. , a linearly independent spanning set. 1. For N and (gi )N the second part, let (ei )N i=1 be orthonormal bases for H .

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