Download Analysis and Probability Wavelets, Signals, Fractals by Palle E. T. Jorgensen, B. Treadway PDF

By Palle E. T. Jorgensen, B. Treadway

Combines research and instruments from probability, harmonic research, operator idea, and engineering (signal/image processing)

Interdisciplinary focus with hands-on technique, beneficiant motivation and new pedagogical techniques

Numerous workouts strengthen basic thoughts and hone computational skills

Separate sections clarify engineering phrases to mathematicians and operator concept to engineers

Fills a spot within the literature

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Additional resources for Analysis and Probability Wavelets, Signals, Fractals

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To reduce the apparent confusion created by the same concept having up to four different names, the glossary includes informal explanations spelling out the reasons behind the differences in current terminology from neighboring fields. hi item (i) we attempt to translate the various engineering terms and constructs into mathematical formulas. This is a continuation of a theme we started in an AMS Notices article [Jor03]. For (iii), we apologize for the subjective nature of our comments; and we readily acknowledge the diflficulties in writing history of events that are rather modem on a mathematical scale.

In. 0)}) = W {zi,x) --W {n, . . ,,^x ({0}) . 13). 3. 1) which may be understood with the use of transition probabilities for a certain random-walk model: (a) (p{t) = (p (2t) + ^ (2/ - 1), Haar's wavelet. 2(a); (b) (p{t) = (p {It) + (p{2t - 3), the stretched Haar wavelet. 2(b); (c) (p{t) = cp (30 -\-(pOt — 2), Cantor's example; and (d) ^ ( 0 = ^ ^ ( 2 0 + ^ ^ ( 2 / - 1) + ^ ^ ( 2 / ~ 2) + ^-^(p{2t - 3), Daubechies' scaling fimction. We shall be interested in solutions (p, called scaling functions, to (a)-(d) which satisfy the fiirther normalization I (p{t)dt = \.

Magically, and by hindsight, these filter systems serve at the same time to give us orthogonal wavelets, subject to a technical condition discussed in Chapter 5. ) Wavelet algorithm in the dyadic case: ¥„ = Vn+\ + ^w+i, w = 0, 1, Wavelet algorithm in the 7V-adic case: Vn-\ = Vn -\- Wn, Wn = WP + WP + .. • + wP-^\ n = l,2,.... , mjsi-i A decreasing family of resolution subspaces FQ D Fi D F2 D • •, iV-adic case, N = 4, Vn-\ = F„ + Wn. Wn = wP + wP + wP, n = l,2,.. A selection of a dyadic wavelet packet.

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