Download An Introduction to Frames and Riesz Bases by Ole Christensen PDF

By Ole Christensen

ISBN-10: 1461265002

ISBN-13: 9781461265009

The idea for frames and bases has constructed speedily in recent times as a result of its position as a mathematical software in sign and snapshot processing. during this self-contained paintings, frames and Riesz bases are awarded from a practical analytic perspective, emphasizing their mathematical homes. this can be the 1st complete e-book to target the overall houses and interaction of frames and Riesz bases, and hence fills a spot within the literature.
Key features:
* easy effects provided in an obtainable approach for either natural and utilized mathematicians
* huge workouts make the paintings compatible as a textbook to be used in graduate courses
* complete proofs incorporated in introductory chapters; simply easy wisdom of practical research required
* particular structures of frames with purposes and connections to time-frequency research, wavelets, and nonharmonic Fourier series
* chosen examine issues provided with options for extra complicated issues and additional reading
* Open difficulties to simulate additional research

An advent to Frames and Riesz Basis may be of curiosity to graduate scholars and researchers operating in natural and utilized arithmetic, mathematical physics, and engineering. pros operating in electronic sign processing who desire to comprehend the idea at the back of many glossy sign processing instruments can also locate this e-book an invaluable self-study reference.

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Additional info for An Introduction to Frames and Riesz Bases

Example text

N. 19) constitute an orthonormal basis for en . Proof. Since {ek}k=l are n vectors in an n-dimensional vector space, it is enough to prove that they constitute an orthonormal system. It is clear 20 that 1. Frames in Finite-dimensional Inner Product Spaces llekll = 1 for all k. £, (ek, ee )= -n1 L e n . k-l . t-l 2rrt(J-1)- -2rrt(J-1)n e n j=1 Using the formula (1- x)(1 we get + x + · · · + xn- 1 ) 1 = -n Le n- 1 · · k-t 21rt)n . j=O = 1- xn with x = e2 "i k;;-t, 0 The basis {ek}k= 1 is called the discrete Fourier transform basis.

Consider an arbitrary subset { k1 , k2 , ... , kn} <; { 1, 2, ... , m}. ' we obtain (-h1-l 1 -h2- 1 1 tri e 2tri (kJ-1)(n-1) e2 . (kn-1)(n-1) e 2 rrt 1 (~ T1l z(k 1 -1)(n-1) z(k 2 -1)(n-1) z(kn-1)(n-1) ; this is a Vandermonde matrix with determinant 1 mn/2 IT n (zk,-1- zk,-1) =/- 0. 1. 5 23 Pseudo-inverses and the singular value decomposition It is well known from linear algebra that not all matrices have an inverse. Keeping in mind how useful inverses are, it is natural to search for some types of "generalized inverses" in case no inverse exists, which capture at least some of the nice properties.

Xk =f. Aj for k =f. j. kx} k=l as functions on I. kx}k=l are linearly independent. Proof. kx hEZ are linearly independent as functions on any bounded interval]a, b[, where a, bE IR, 30 a 1. Frames in Finite-dimensional Inner Product Spaces < b. kx = 0, Vx E]a, b[. k(x+~) =0, V E]a-b b-a[ ~ 2 ' X k=1 ~ ·>. ~dke' kx k=1 = 0, . >. k)Je' kx k=1 = 0, . a -b b-a Vx E]2-, - 2-[. By differentiating this equation j times, j ~ 2 = 0, 1, · · ·, we obtain that a-b b-a Vx E]2 -, - - [ , j 2 = 0, 1, · · · . n-1 = 0, ...

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