The Fourier Series

We saw in Section 4.2 that the plucked string supports certain spatial vibrations, called modes. These modes have a very special relationship in the case of the plucked string (and some other limited systems) in that their frequencies are all integer multiples of one basic sinusoid, called thefundamental. This special series of sinusoids is called a harmonic series, and lies at the basis of the Fourier series representation of shapes, waveforms, oscillations, etc. The Fourier series solves many types of problems, including physical problems with boundary constraints, but is also applicable to any shape or function. Any periodic waveform (repeating over and over again), can be transformed into a Fourier series, written as  

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We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

The Fourier Series, Fourier Transform and Fast Fourier Transform... 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

This form of the Fourier series is entirely equivalent to the first form and the an and bn ean be obtained from the Cn and viee versa. For example, the en amplitudes are obtained by... 

If the transverse load is but one term of the Fourier series, that is. 

In the Fourier space, the order parameter 0(r) for periodic structures is approximated by the Fourier series... 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

Though he won the prize he did not win the outright acclaim of his referees. They accepted that Fourier had formulated heat flow correctly but felt that his methods were not without their difficulties. The use of the Fourier Series was still controversial. It was only when he had returned to Paris for good (around 1818) that he could get his work published in his seminal book, The Analycical Theoiy of Heat. 

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

A similar expression applies for b . The generalization for a function /(9) with a finite number of finite discontinuities is straightforward. At an angle 9q of discontinuity, the Fourier series converges to a value of /(9) mid-way between the left and right values... 

We now investigate the relation between the average of the square of /(0) and the coefficients in the Fourier series for /(0). For this purpose we select the Fourier series (B.8), although any of the other expansions would serve as well. In this case the average of /(0)p over the interval -Ji 0 s= jr is... 

Section 3.4. If some of the terms in the Fourier series are missing, so that the set of basis functions in the expansion is incomplete, then the corresponding coefficients on the right-hand side of equation (B.16) will also be missing and the equality will not hold. 

Here we are interested in escape out of the domain L specified by a single cycle of the potential that is out of a domain of length n that is the domain of the well. Because the bistable potential of Eq. (5.42) has a maximum at x = n/2 and minima at x = 0, x = 7t, it will be convenient to take our domain as the interval —7t/2 < x < n/2. Thus we will impose absorbing boundaries at x = —n/2, x = n/2. Next we shall impose a second condition that all particles are initially located at the bottom of the potential well so that x0 = 0. The first boundary condition (absorbing barriers at —n/2, n/2) implies that only odd terms in p in the Fourier series will contribute to Y (x). While the second ensures that only the cosine terms in the series will contribute because there is a null set of initial values for the sine terms. Hence... 

If in a more general formulation the variable nx/l is replaced by x, the range changes to 0, n and the Fourier series becomes... 

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function /(x) defined over the interval [ —X, X] is a convenient example... 

Changing x to — x in the first integral and recognizing that cos is an even function shows that b is zero for all n. The Fourier series expansion of the boxcar function is therefore... 

Find the Fourier series expansion of the ramp function (Figure 2.11) defined over the interval [—X,X] by... 

It will be left as an exercise for the reader to show that the cosine terms bn, including the mean value b0, are zero. The Fourier series expansion of the ramp function is therefore... 

The major shortcoming of the spectral method is the rate of convergence. Its ability to resolve eigenvalues is restricted by the width of the filter, which in turn is inversely proportional to the length of the Fourier series (the uncertainty principle). Thus, to accurately characterize an eigenpair in a dense spectrum, one might have to use a very long Chebyshev recursion. 

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