Fourier polynomial series

The Fourier polynomial series is not a sequence of increasing powers of x. like the Chebychev polynomial, but a series of sines and cosines of increasing frequency. In fact, there is no longer a notion of x and y, as in the initial example of polynomial approximation, but just y. a series of num-... 

We may solve for the electron distribution function by expanding it in Legendre polynomials in cos 6 (where v = (v,6,Fourier series in cot we shall use here only the first-order terms ... 

A Fourier series is an example of an orthogonal polynomial, meaning that the individual terms which it comprises are independent of each other. It should be possible, therefore, to dissect a complex rotational energy profile into a series of N-fold components, and interpret each of these components independent of all others. For example, the onefold term (the difference between syn and anti conformers) in /7-butane probably reflects the crowding of methyl groups. 

Notice that we have approximated a discontinuous function by a continuous one. It turns out that any function in L —1, 1] can be approximated by trigonometric polynomials — this is one of the important results of the theory of Fourier series. ... 

Having obtained a fitting function y x) in the form of a polynomial, Fourier series or integral transform, or other form, we may differentiate it or integrate it as desired. 

However, the expansion above is still impractical for our purposes, because the functions As(q), Bs(q),. .. still need to be expanded in an infinite Fourier series of the angles e.g., we should write A q) = J2kez akexp(i(k,qj). It is more convenient to work with trigonometric polynomials, so that every part of the expansion contains only a finite number of terms. To this end, we introduce a Fourier cutoff by splitting every function of the angles in an infinite number of slices that contain only a finite number of Fourier modes. This may be done in many arbitrary ways, so let us illustrate just one method. We choose an arbitray integer K, ad write, e.g.,... 

Instead of expanding this function in Fourier series in S (the Laplace approach) or power series in g (Legendre polynomials), a best-fit approach is used. We write... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

Choice of Basis Function Set (Fourier Series and Chebyshev Polynomials)... 

Spectral simulation methods have been used since the late 1960s. The books by Gottlieb and Orszag [132], Canuto et al. [133], and Boyd [134] discuss spectral methods. Typically, one expands the velocity field and other dependent variables in Fourier series or other functions such as Chebyshev polynomials that are solutions of second order Sturm-Liouville problems. 

The pressure P can be expanded with respect to Y in several ways and also different sets of weighting functions for the Reynolds equation can be used. The choice as indicated in 4.2 seems a natural one. But with this set of basis and weighting functions, and also with other sets of polynomial functions, the resulting matrix appears to be very ill-conditioned for larger sets (N>5). For this reason Fourier series are used, so P is approximated by... 

Naim and Liu [96] used a Bessel-Fourier series stress function and added polynomial terms to provide a nearly exact solution to the stress transfer fixim the matrix to a fragmented fiber through an imperfect interphase. This solution satisfies equilibrium and compatibility every place and satisfies exactly most boundary conditions with the exception of the fiber axial stress. They also proposed the use of an interphase parameter, Ds, and provide a physical interpretation as ... 

It can be more useful to use spherical co-ordinates [3] (see FIGURE 2). Any cylindrically symmetrical function with a centre of inversion can be developed into a series of even-order Legendre polynomials P2n, in a fashion similar to the way a periodic function is depicted by a Fourier series. The scattered intensity distribution over the reciprocal space can thus be represented as... 

The latter half of the book is devoted to those areas of mathematics normally not covered in the prerequisite calculus courses taken for physical chemistry. A number of chapters have been expanded to include material not found in the first edition, but again, for the most part, at the introductory level. For example, the chapter on differential equations expands on the series method of solving differential equations and includes sections on Hermite, Legendre, and Laguerre polynomials the chapter on infinite series includes a section on Fourier transforms and Fourier series, in rtant today in many areas of spectroscopy and the chapter on matrices and determinants includes a section on putting matrices in diagonal form, a major type of problem encountered in quantum mechanics. 

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