Download Applied Mathematics: Data Compression, Spectral Methods, by Charles K. Chui PDF

By Charles K. Chui

This textbook, except introducing the elemental elements of utilized arithmetic, makes a speciality of contemporary issues similar to info facts manipulation, details coding, facts approximation, information dimensionality aid, info compression, time-frequency and time scale bases, picture manipulation, and photograph noise elimination. The tools taken care of in additional element comprise spectral illustration and “frequency” of the knowledge, offering helpful details for, e.g. information compression and noise elimination. in addition, a unique emphasis is usually wear the concept that of “wavelets” in reference to the “multi-scale” constitution of data-sets. The presentation of the booklet is straight forward and simply obtainable, requiring just some wisdom of easy linear algebra and calculus. All very important techniques are illustrated with examples, and every part comprises among 10 an 25 routines. A educating consultant, looking on the extent and self-discipline of directions is integrated for lecture room instructing and self-study.

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Extra info for Applied Mathematics: Data Compression, Spectral Methods, Fourier Analysis, Wavelets, and Applications

Sample text

Where the value 1 occurs at the kth entry, while all the other entries are equal to zero. It is clear that each delta sequence δ k is in p for any p with 0 ≤ p ≤ ∞. In the following, we will see that while S = {δ k : k = 0, ±1, ±2, . } is an algebraic basis of 0 , it is not an algebraic basis of p for 0 < p ≤ ∞. Example 2 Verify that S = {δ k : k = 0, ±1, ±2, · · · } is an algebraic basis for 0. Solution Indeed, for any x ∈ 0 , there are only finitely many entries xk of x that are nonzero. Thus, we have xk = 0 for all |k| > K for some positive integer K .

Clearly this representation for x is unique, so that S is a basis for 2. In addition, for j, k = 0, ±1, ±2, . , ∞ δk , δ j = δk−m δ j−m = δ j−k , m=−∞ and hence, S is an orthonormal basis for 2. Next, we introduce the concept of orthogonal projection. Let w1 , . . , wn be nonzero orthogonal vectors in an inner-product √ ssspace V over some scalar field x, x , x ∈ V, and consider the algeF, with inner product ·, · and norm x = braic span W = span{w1 , . . , wn }. For any v ∈ V, the vector Pv ∈ W, defined by Pv = c1 (v)w1 + c2 (v)w2 + · · · + cn (v)wn , with c j (v) = v, w j , wj 2 j = 1, 2, .

To derive an explicit formula of dist(v, W), we rely on the orthogonal basis w1 , . . 4 Bases of Sequence and Function Spaces 45 n v − Pv 2 n = v− c j (v)w j , v − j=1 c j (v)w j j=1 n n = v, v − c j (v) v, w j − j=1 n = v 2 − j=1 n = v 2 − j=1 n c j (v) w j , v + j=1 | v, w j |2 − wj 2 n j=1 c j (v) 2 wj, wj j=1 | v, w j |2 + wj 2 n j=1 | v, w j |2 wj 2 |2 | v, w j . 8) Let us summarize this result in the following theorem. Theorem 3 Bessel s inequality Let W be any subspace of an inner-product space V with orthogonal basis w1 , .

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