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

By Ole Christensen

This revised and extended monograph offers the overall concept for frames and Riesz bases in Hilbert areas in addition to its concrete realizations inside of Gabor research, wavelet research, and generalized shift-invariant platforms. in comparison with the 1st variation, extra emphasis is wear particular structures with beautiful homes. in accordance with the exiting improvement of body concept over the past decade, this moment version now comprises new sections at the quickly growing to be fields of LCA teams, generalized shift-invariant structures, duality concept for besides Gabor frames as wavelet frames, and open difficulties within the field.

Key gains include:

*Elementary advent to border thought in finite-dimensional spaces
* uncomplicated effects provided in an obtainable means for either natural and utilized mathematicians
* vast routines make the paintings compatible as a textbook to be used in graduate courses
* complete proofs includ

ed in introductory chapters; in basic terms uncomplicated wisdom of practical research required
* specific 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 when it comes to Gabor structures at the straightforward teams; connections to sampling theory

* chosen learn issues awarded with suggestions for extra complex subject matters and extra readin

g

* 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. pros operating in electronic sign processing who desire to comprehend the idea at the back of many sleek sign processing instruments can also locate this e-book an invaluable self-study reference.

Review of the 1st edition:

"Ole Christensen’s An creation to Frames and Riesz Bases is a primary advent to the sphere … . The booklet presents a very good exposition of those subject matters. the fabric is huge adequate to pique the curiosity of many readers, the integrated routines provide a few attention-grabbing demanding situations, and the assurance offers sufficient history for these new to the topic to start carrying out unique research."

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

Show description

Read Online or Download An Introduction to Frames and Riesz Bases PDF

Similar functional analysis books

Real Functions - Current Topics

So much books dedicated to the idea of the indispensable have missed the nonabsolute integrals, even though the magazine literature when it comes to those has turn into richer and richer. the purpose of this monograph is to fill this hole, to accomplish a examine at the huge variety of sessions of genuine services which were brought during this context, and to demonstrate them with many examples.

Analysis, geometry and topology of elliptic operators

Glossy thought of elliptic operators, or just elliptic thought, has been formed by way of the Atiyah-Singer Index Theorem created forty years in the past. Reviewing elliptic concept over a extensive diversity, 32 top scientists from 14 various international locations current fresh advancements in topology; warmth kernel concepts; spectral invariants and slicing and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.

Introduction to complex analysis

This e-book describes a classical introductory a part of advanced research for college scholars within the sciences and engineering and will function a textual content or reference publication. It areas emphasis on rigorous proofs, offering the topic as a primary mathematical thought. the amount starts with an issue facing curves on the topic of Cauchy's crucial theorem.

Extra info for An Introduction to Frames and Riesz Bases

Sample text

M are called frame coefficients. 3; it is called the canonical dual frame of {fk }m k=1 . 6 Let {ek }2k=1 be an orthonormal basis for a two-dimensional vector space V with inner product. Let f1 = e1 , f2 = e1 − e2 , f3 = e1 + e2 . Then {fk }3k=1 is a frame for V . Using the definition of the frame operator, 3 f, fk fk , Sf = k=1 we obtain that Se1 = e1 + e1 − e2 + e1 + e2 = 3e1 Se2 = and −(e1 − e2 ) + e1 + e2 = 2e2 . Thus, 1 1 e1 , S −1 e2 = e2 . 3 2 By linearity, the canonical dual frame is S −1 e1 = {S −1 fk }3k=1 = = {S −1 e1 , S −1 e1 − S −1 e2 , S −1 e1 + S −1 e2 } 1 1 1 1 1 { e1 , e1 − e2 , e1 + e2 }.

Using the definition of the frame operator, 3 f, fk fk , Sf = k=1 we obtain that Se1 = e1 + e1 − e2 + e1 + e2 = 3e1 Se2 = and −(e1 − e2 ) + e1 + e2 = 2e2 . Thus, 1 1 e1 , S −1 e2 = e2 . 3 2 By linearity, the canonical dual frame is S −1 e1 = {S −1 fk }3k=1 = = {S −1 e1 , S −1 e1 − S −1 e2 , S −1 e1 + S −1 e2 } 1 1 1 1 1 { e1 , e1 − e2 , e1 + e2 }. 5, the representation of f ∈ V in terms of the frame is given by 3 f f, S −1 fk fk = k=1 = 1 1 1 1 1 f, e1 e1 + f, e1 − e2 (e1 − e2 ) + f, e1 + e2 (e1 + e2 ).

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 . The jkth entry in the matrix representation is U ek , ej .

Download PDF sample

Rated 4.51 of 5 – based on 14 votes