An Introduction to Frames and Riesz Bases by Ole Christensen

By Ole Christensen

This revised and improved monograph offers the overall idea for frames and Riesz bases in Hilbert areas in addition to its concrete realizations inside of Gabor research, wavelet research, and generalized shift-invariant structures. in comparison with the 1st version, extra emphasis is wear particular buildings with beautiful homes. in accordance with the exiting improvement of body conception during the last decade, this moment version now contains new sections at the quickly transforming into fields of LCA teams, generalized shift-invariant platforms, duality conception for in addition Gabor frames as wavelet frames, and open difficulties within the field.

Key positive factors include:

*Elementary advent to border thought in finite-dimensional spaces
* uncomplicated effects awarded 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 includ

ed in introductory chapters; in simple terms easy wisdom of sensible research required
* particular structures of frames and twin pairs of frames, with purposes and connections to time-frequency research, wavelets, and generalized shift-invariant systems

* dialogue of frames on LCA teams and the concrete realizations by way of Gabor structures at the ordinary teams; connections to sampling theory

* chosen learn issues offered with concepts for extra complicated subject matters and extra readin


* Open difficulties to stimulate extra research

An creation to Frames and Riesz Bases will be of curiosity to graduate scholars and researchers operating in natural and utilized arithmetic, mathematical physics, and engineering. execs operating in electronic sign processing who desire to comprehend the idea in the back of many smooth sign processing instruments can also locate this booklet an invaluable self-study reference.

Review of the 1st edition:

"Ole Christensen’s An creation to Frames and Riesz Bases is a chief creation to the sphere … . The ebook offers a superb exposition of those themes. the fabric is vast adequate to pique the curiosity of many readers, the incorporated workouts offer a few fascinating demanding situations, and the assurance offers sufficient historical past for these new to the topic to start undertaking unique research."

— Eric S. Weber, American Mathematical per month, Vol. 112, February, 2005

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2 k=1 So λmin is a lower frame bound, and λmax is an upper frame bound. That they are the optimal frame bounds follows by taking f to be an eigenvector corresponding to λmin (respectively λmax ). For the proof of (ii), we have n n k=1 n λk ||ek ||2 λk = = k=1 Sek , ek k=1 n m | e k , f |2 . = k=1 =1 Interchanging the sums and using that {ek }nk=1 is an orthonormal basis for V now gives (ii). For the proof of (iii), the assumptions imply that the set of eigenvalues {λk }nk=1 consists of the frame bound A repeated n times; thus, the result follows from (ii).

V ∗V D−1 F U ∗f This means that we can use the coefficients {ck }m k=1 = V D−1 F U ∗f for the reconstruction of f , regardless how the entries in the matrix F are chosen. 5, is optimal in the sense that the 2 -norm of the coefficients is minimized. However, there are many cases where other properties than minimal 2 -norm are more relevant. The matrix V D−1 F U∗ is frequently called a generalized inverse of T . 7 Finite-Dimensional Function Spaces The rest of the book will deal with frames in infinite-dimensional vector spaces, with concrete constructions in function spaces like L2 (−π, π) and L2 (R); the exact definition of these spaces will be given in Chapter 2, and for the moment we simply consider L2 (I), I ⊆ R as the set of functions for which |f (x)|2 dx < ∞.

We also remind the reader that any basis can be turned into an orthonormal basis by applying the Gram–Schmidt orthogonalization procedure. We now turn to a discussion of frames for Cn . Note that we consequently identify operators U : Cn → Cm with their matrix representations with respect to the canonical bases in Cn and Cm . Letting {ek }nk=1 denote the canonical orthonormal basis in Cn and {ek }m k=1 the canonical orthonormal basis in Cm , the matrix representation of U is the m × n matrix, where the kth column consists of the coordinates of the image under U of the kth basis vector in U , in terms of the given basis in W .

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