Convergence of Fourier Series

If/(x) is piecewise continuous and periodic with period 2it, then 

The following theorem can be used to establish point convergence of a given Fourier series [1,4]. 

Iff(x) is piecewise smooth and periodic with period 2ir in —it x it, then 

Proof of Theorem 5.1 can be found elsewhere [1], however, an example of its 

Another useful result regarding convergence is given below. This result is known as the Riemann-Lebesgue lemma. 

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Vol. 1785 J. Arias de Reyna, Pointwise Convergence of Fourier Series (2002)... 

It is a necessary condition for the convergence of Fourier series that the coefficients become smaller and smaller and approach zero as n becomes larger and larger. If convergence is fairly rapid, it might be possible to approximate a Fourier series by a partial sum. Figure 11.2 shows three different partial sums of the Fourier sine series that represents the square-wave function... 

More recently, as quoted previously, Delhalle and Harris (13) have provided a general analysis of the convergence of Dp in terms of basic theorems on the convergence of Fourier series coefficients. They show that the convergence of Dp is essentially determined by the analytic properties of D (k)... 

In this section no attempt has been made to discuss the very difficult problem of the convergence of Fourier-Bcssel series. For a very full discussion of this topic the reader is referred to Chapter XVIII of G. N. Watson s A Trealise on the Theory of Bessel Functions, 2nd. edit., (Cambridge University Press, 1044). 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

These results of Bruns have been supplemented by Poincare s investigations 1 these lead to the following conditions Apart from special cases, it is not possible to represent strictly the motion of the perturbed system by means of convergent /-fold Fourier series in the time and magnitudes Jk constant in time, which could serve for the fixation of the quantum states. For this reason it has hitherto been impossible to carry out the long-sought-for proof of the stability of the planetary system, i.e. to prove that the distances of the planets from one another and from the sun remain always within definite finite limits, even in the course of infinitely long periods of time. 

An adequate discussion of the conditions of convergency of Fourier s series must be omitted. W. E. Byerly s An Elementary Treatise on Fourier s Series, etc., is one of the best practical works on the use of Fourier s integrals in mathematical physics. J. Fourier s pioneer work Thiorie analytiquc de la Chaleur, Paris, 1822, is perhaps as modern as any other work on this subject see also W. Williams, Phil. Mag. [5], 42, i25,1896 Lord Kelvin s Collected Papers and Riemann-Weber s work (Z.c.), etc. 

One of the most common solution techniques applicable to linear homogeneous partial differential equation problems involves the use of Fourier series. A discussion of the methods of solution of linear partial differential equations will be the topic of the next chapter. In this chapter, a brief outline of Fourier series is given. The primary concerns in this chapter are to determine when a function has a Fourier series expansion and then, does the series converge to the function for which the expansion was assumed Also, the topic of Fourier transforms will be briefly introduced, as it can also provide an alternative approach to solve certain types of linear partial differential equations. 

In case the curve y = fix) is symmetrical with respect to the origin, the a s are all zero, and the series is a sine series. In case the curve is symmetrical with respect to the y axis, the fc s are all zero, and a cosine series results. (In this case, the series will be valid not only for values of x between — c and c, but also for x = — c and x = c.) A Fourier series can always be integrated term by term but the result of differentiating term by term may not be a convergent series. 

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... 

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. 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

If the Fourier synthesis is carried out by adding in the strong reflections first, we will see how fast the Fourier series converges to the projected potential. The positive potential contribution from the reflection is shown in white, whereas the negative potential contribution is shown in black. Most of the atoms are located in the white regions of each cosine wave, but the exact atomic positions will not become evident until a sufficiently large number of structure factors have been added up. 

Convergence of the actual solution to the self-similar one over time occurs in a way similar to convergence of a Fourier series to a discontinuous function as the number of terms in the sum increases the well-known Gibbs phenomenon leads to the fact that, near the discontinuity, for any number of terms, the maximum difference between the series and the function does not approach zero however, the width of the region in which the series differs noticeably from the function approaches zero as the number of terms increases. 

The spatial structure of the turbulent field is contained in the eigenvectors. We see that the eigenvectors are expanded as a Fourier series in the directions of flow homogeneity, x and y. The rate of convergence of the sequence of eigenvalues is a sensitive indicator of the presence and relative importance of coherent structures. EOF analysis provides not only an objective measure of the existence of dominant, spatially-extensive structure but, with minimal additional assumptions, allows us to deduce the 3-dimensional structure of the dominant eddies in their mature phase. 

We calculate %(a>) by converting the solution of the fractional diffusion Eq. (55) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,62]. By expanding the distribution function W(< ), f) in Fourier series... 

Equation 2-39 cannot be readily solved as it is written. Osborne Reynolds obtained a solution for it in the form of a Fourier series which converges for eccentricity ratios less than 0.5 and is therefore useful only for lightly loaded bearings. The exact solution of A. Sommerfeld, details of which can be found in most of the standard texts on hydrodynamic lubrication of bearings, yields the following expression ... 

A mathematical series is a sum of terms. A series can have a finite number of terms or can have an infinite number of terms. If a series has an infinite number of terms, an important question is whether it approaches a finite limit as more and more terms of the series are included (in which case we say that it converges) or whether it becomes infinite in magnitude or oscillates endlessly (in which case we say that it diverges). A constant series has terms that are constants, so that it equals a constant if it converges. Afunctional series has terms that are functions of one or more independent variables, so that the series is a function of the same independent variables if it converges. Each term of a functional series contains a constant coefficient that multiplies a function from a set of basis functions. The process of constructing a functional series to represent a specific function is the process of determining the coefficients. We discuss two common types of functional series, power series and Fourier series. 

If we want to produce a series that will converge rapidly, so that we can approximate it fairly well with a partial sum containing only a few terms, it is good to choose basis functions that have as much as possible in common with the function to be represented. The basis functions in Fourier series are sine and cosine functions, which are periodic functions. Fourier series are used to represent periodic functions. A Fourier series that represents a periodic function of period 2L is... 

There are some important mathematical questions about Fourier series, including the convergence of a Fourier series and the completeness of the basis functions. A set of basis functions is said to be complete for representation of a set of functions if a series in these functions can accurately represent any function from the set. We do not discuss the mathematics, but state the facts that were proved by Fourier (1) any Fourier series in is uniformly convergent for all real values of v (2) the set of sine and cosine basis functions in Eq. (6.39) is a complete set for the representation of periodic functions of period 2L. In many cases of functional series, the completeness of the set of basis functions has not been proved, but most people assume completeness and proceed. 

A function does not have to be analytic, or even continuous, in order to be represented by a Fourier series. It is only necessary that the function be integrable. As mentioned in Chapter 5, an integrable function can have step discontinuities, as long as the step in the function is finite. At a step discontinuity, a Fourier series will converge to a value halfway between the value just to the right of the discontinuity and the value just to the left of the discontinuity. 

More rapid convergence of the Fourier series can be achieved if one uses a more efficient damping factor, g , rather than a simple exponential damping, e nt,... 

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