Download Modern Sampling Theory: Mathematics and Applications by John J. Benedetto, Paulo J.S.G. Ferreira PDF

By John J. Benedetto, Paulo J.S.G. Ferreira

Sampling is a primary subject within the engineering and actual sciences. This new edited e-book makes a speciality of fresh mathematical equipment and theoretical advancements, in addition to a few present primary functions of the Classical Sampling Theorem. The Classical Sampling Theorem, which originated within the nineteenth century, is usually linked to the names of Shannon, Kotelnikov, and Whittaker; and one of many good points of this publication is an English translation of the pioneering paintings within the Nineteen Thirties via Kotelnikov, a Russian engineer. Following a technical review and Kotelnikov's article, the ebook contains a huge and coherent variety of mathematical rules crucial for contemporary sampling ideas. those rules contain wavelets and frames, complicated and summary harmonic research, the short Fourier rework (FFT),and specified services and eigenfunction expansions. a number of the functions addressed are tomography and clinical imaging. Topics:. family among wavelet conception, the uncertainty precept, and sampling; . Multidimensional non-uniform sampling idea and algorithms;. The research of oscillatory habit via sampling;. Sampling innovations in deconvolution;. The FFT for non-uniformly disbursed information; . filter out layout and sampling; . Sampling of noisy information for sign reconstruction;. Finite dimensional versions for oversampled clear out banks; . Sampling difficulties in MRI. Engineers and mathematicians operating in wavelets, sign processing, and harmonic research, in addition to scientists and engineers engaged on functions as diverse as clinical imaging and artificial aperture radar, will locate the booklet to be a contemporary and authoritative advisor to sampling theory.

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Extra resources for Modern Sampling Theory: Mathematics and Applications

Example text

A straightforward proof in terms of frames is given in [Ben92, page 465]. ) In the second category, the book by Marvasti [Mar87] presents a comprehensive list of references, and commentary on them, up to 1987. Since then the applications and references have expanded manyfold. We shall resist presenting the litany we know, to save space as well as to save the embarrassment of showing what we don't know, see [MarOO], [BWa]. 3 Outline of the Book Chapter 2 is Victor E. Katsnelson's translation for this book of the classical paper by Kotel'nikov.

Is the inner product in H. Let k(t, u) := (kt, ku) for t, u E T. Clearly, kt (u) = k( t, u); k( t, u) is called the reproducing kernel (RK) of H. We denote by Hk the RKHS with RK k(t, u) which is Hermitian and positive definite. , the Paley-Wiener space B1C of bandlimited functions, has a number of special properties. It is a reproducing kernel Hilbert space with reproducing kernel k( t, u ) =sin7r(t-u) ( ). 7rt-u The function k( t, u) has zeros at t = m, u = n, where m, nEZ, m =I- n; and Sn(t) = k(t, n) Chapter 3.

22) of Paley and Wiener. Landau's work in [Lan67a], [Lan67b] was influenced by Beurling's ideas, and some of Beurling's results can be expressed in terms of Fourier frames. 20 Benedetto and Ferreira For example, Beurling proved that if A ~ i is uniformly discrete and p = sup dist(r, A) "lEi (dist (r, A) is the Euclidean distance between 'Y and A), then the condition Rp < implies {e A ; A E A} is a Fourier frame for L 2 [ - R, R], see [Beu60], [Beu66a], [Beu89]. , with the approach in Chapter 16.

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