Charles K. Chui's Applied Mathematics: Data Compression, Spectral Methods, PDF

By Charles K. Chui

This textbook, except introducing the fundamental points of utilized arithmetic, specializes in contemporary subject matters resembling info info manipulation, details coding, facts approximation, facts dimensionality relief, information compression, time-frequency and time scale bases, snapshot manipulation, and photograph noise elimination. The equipment handled in additional aspect comprise spectral illustration and “frequency” of the information, delivering necessary details for, e.g. facts compression and noise elimination. additionally, a unique emphasis can also be wear the concept that of “wavelets” in reference to the “multi-scale” constitution of data-sets. The presentation of the ebook is simple and simply available, requiring just some wisdom of uncomplicated linear algebra and calculus. All vital thoughts are illustrated with examples, and every part comprises among 10 an 25 workouts. A educating consultant, reckoning on the extent and self-discipline of directions is incorporated for lecture room educating and self-study.

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Additional resources for Applied Mathematics: Data Compression, Spectral Methods, Fourier Analysis, Wavelets, and Applications

Example text

Wn be any orthogonal basis of W. Later in this section, we will derive a procedure, called the Gram-Schmidt process, for computing an orthonormal basis of W from any given basis of W. Let w ∈ W be arbitrarily chosen. Then by (c) of Theorem 1, since Pv − w ∈ W, the vector v − Pv is orthogonal to Pv − w. Thus, it follows from the Pythagorean theorem (see Theorem 4 in Sect. 24) that v−w 2 = (v − Pv) + (Pv − w) 2 = v − Pv 2 + Pv − w 2 , so that v − Pv 2 ≤ v−w 2 for all w ∈ W. The non-negative real number minw∈W v − w is called the distance of v from W, denoted by dist(v, W).

6) is a vector space over the scalar field C. This statement remains valid if Cm,n and C are replaced by Rm,n and R, respectively. Definition 2 Subspaces A nonempty subset W of a vector space V over F is called a subspace of V, if W is also a vector space over F with the same addition and scalar multiplication operations of V. Since W ⊂ V, the operations in (i)–(ii) and (iv)–(vii) of Definition 1 are already valid for vectors in W. In addition, since W is nonempty, there exists v0 ∈ W. Thus, from (v) in Definition 1, we see 0 = 0v0 ∈ W.

Now, it is clear from Fig. 2 that a b f (x)d x + 0 f −1 (y)dy ≥ ab, 0 where equality holds if and only if b = f (a) = a p−1 . By direct calculation, we have a a f (x)d x = 0 x p−1 d x = 0 and b 0 b f −1 (y)dy = 0 1 1 y p−1 dy = ap , p b p−1 1 p−1 +1 +1 = bq , q 18 1 Linear Spaces since 1 p 1 +1= = p−1 p−1 1− = q. 1 p This completes the proof of Theorem 3. We are now ready to state and prove the following inequality of Hölder. 10). Then for any two sequences x = {x j } ∈ p and y = {y j } ∈ q , the sequence {x j y j } is a sequence in 1 , and |x j y j | ≤ |x j | p j 1/ p |y j |q j 1/q .

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