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

By Charles K. Chui

This textbook, except introducing the fundamental points of utilized arithmetic, specializes in fresh themes similar to info facts manipulation, details coding, information approximation, facts dimensionality aid, info compression, time-frequency and time scale bases, photo manipulation, and photograph noise removing. The tools handled in additional element contain spectral illustration and “frequency” of the knowledge, offering useful details for, e.g. info compression and noise elimination. moreover, 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 booklet is uncomplicated and simply available, requiring just some wisdom of undemanding linear algebra and calculus. All vital suggestions are illustrated with examples, and every part includes among 10 an 25 workouts. A educating advisor, reckoning on the extent and self-discipline of directions is integrated for lecture room instructing and self-study.

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**Additional info for Applied Mathematics: Data Compression, Spectral Methods, Fourier Analysis, Wavelets, and Applications (Mathematics Textbooks for Science and Engineering)**

**Example text**

J=−K 38 1 Linear Spaces K To show that the above expression is unique, let x = dk δ k , and take the differj=−K ence of these two sums to arrive at K (xk − dk )δ k = x − x = 0. j=−K That is, (. . , 0, x−K − d−K , . . , x K − d K , 0, . ) = 0 = (. . , 0, 0, . . , 0, 0, . ), so that xk = dk , −K ≤ k ≤ K . Therefore, every x ∈ 0 can be written uniquely as a finite linear combination of S = {δ k : k = 0, ±1, ±2, . }. That is, S is an algebraic basis of 0 . Example 3 Give an example to illustrate that S = {δ k : k = 0, ±1, ±2, · · · } is not an algebraic basis of p for 0 < p ≤ ∞.

Solution We have 30 1 Linear Spaces 1 f1 , f2 = f1 2 0 1 = 2 1 | f 1 (x)|2 d x = 1 , 2 1d x = 1, 0 2 = xd x = 0 0 f2 1 f 1 (x) f 2 (x)d x = 1 | f 2 (x)|2 d x = 0 x 2d x = 0 and cos θ = 1 ; 3 √ 3 f1 , f2 = . f1 f2 2 Thus, the angle θ between f 1 and f 2 is θ = cos−1 √ 3 π = . 2 6 For g1 and g2 , we have 1 g1 , g2 = cos πx d x = 0 1 sin πx π 1 = 0. x=0 Therefore g1 and g2 are orthogonal to each other, and the angle between them is π2 . Observe that there is no geometric meaning for the angle between two functions.

8) with these examples. (a) x, y ∈ R3 , with x = (1, 2, −1) and y = (0, −1, 4). (b) x, y ∈ 2 , with x = {x j } and y = {y j }, where x j = y j = 0 for j ≤ 0; 1 1 x j = , y j = (−1) j , for j > 0. j j ∞ ∞ 1 1 π2 = by observing that Hint: . Compute 2 j 6 (2 j − 1)2 j=1 j=1 ∞ j=1 1 = j2 ∞ j=1 1 + (2 j − 1)2 ∞ j=1 1 . (2 j)2 Apply the same trick of writing the summation as the sum over the even indices ∞ (−1) j and the sum over the odd indices to compute . j2 j=1 Exercise 5 In each of the following, compute f, g , f , and g for the space L 2 [0, 1].