Engineering applications of noncommutative harmonic analysis by Chirikjian G.S., Kyatkin A.B.

By Chirikjian G.S., Kyatkin A.B.

The classical Fourier rework is without doubt one of the most generally used mathematical instruments in engineering. besides the fact that, few engineers recognize that extensions of harmonic research to services on teams holds nice strength for fixing difficulties in robotics, picture research, mechanics, and different parts. for those who can be conscious of its capability price, there's nonetheless no position they could flip to for a transparent presentation of the heritage they should practice the idea that to engineering problems.Engineering functions of Noncommutative Harmonic research brings this strong instrument to the engineering international. Written particularly for engineers and laptop scientists, it bargains a pragmatic remedy of harmonic research within the context of specific Lie teams (rotation and Euclidean motion). It offers just a restricted variety of proofs, focusing in its place on supplying a evaluation of the basic mathematical effects unknown to so much engineers and specific discussions of particular applications.Advances in natural arithmetic can result in very tangible advances in engineering, yet provided that they're to be had and available to engineers. Engineering purposes of Noncommutative Harmonic research offers the potential for including this important and powerful strategy to the engineer's toolbox.

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CLASSICAL FOURIER ANALYSIS 31 The DFT of such a function is defined as5 1 fˆk = F fj = N N−1 fj e−i2πj k/N . 35) j =0 This can be viewed as an N −point approximation in the integral defining the Fourier coefficients fˆ(k), where the sample points are xj = j L/N , fj = f (xj ), and dx is approximated as x = L/N . Or it can simply be viewed as the definition of the Fourier transform of a function whose arguments take values in a discrete set. As with the Fourier series and transform, the DFT inherits its most interesting properties from the exponential function.

Using vector notation fˆ(ω) = f (x)e−iω·x dx RN where dx = dx1 . . dxN . The inversion, viewed as N one-dimensional inversions, yields f (x) = where dω = dω1 . . dωN . 3 23 Using the Fourier Transform to Solve PDEs and Integral Equations The Fourier transform is a powerful tool when used to solve, or simplify, linear partial differential equations (PDEs) with constant coefficients. 4. The usefulness of the Fourier transform in this context is a direct result of the operational properties and convolution theorem discussed earlier in this chapter.

Using this property and the definition of the DFT it is easy to show that fˆk+N = fˆk . 36) k=0 by observing that for a geometric sum with r = 1 and |r| ≤ 1, N−1 rk = k=0 1 − rN . 1−r Setting r = ei2π(n−m)/N for n = m yields the property that r N = 1, and so the numerator in the above equation is zero in this case. When n = m, all the exponentials in the sum reduce to the number 1, and so summing N times and dividing by N yields 1. Equipped with Eq. 36), one observes that N−1 fˆk ei2πj k/N = k=0 N−1 k=0 1 N N−1 = fn n=0 N−1 fn e−i2πnk/N ei2πj k/N n=0 1 N N−1 ei2π(j −n)k/N k=0 N−1 = fn δj,n n=0 and thus the DFT inversion formula N−1 fj = fˆk ei2πj k/N .

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