Week 15. Vladimir Dobrushkin


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1 Week 5 Vladimir Dobrushkin 5. Fourier Series In this section, we present a remarkable result that gives the matehmatical explanation for how cellular phones work: Fourier series. It was discovered in the beginning of nineteenth century by Joseph Fourier (768 8) that an arbitrary function f(x) defined on the interval of length can be represented by a convergent trigonometric series: f(x) = a + ( +b n sin nπx ). (5..) n= Eq. (5..) means that at every point x in the interval [,], the function f(x) is the limit of the partial sums: f(x) = lim N S N(x), S N (x) = a N + ( +b n sin nπx ). (5..) n= Note that for some points, the limit (5..) may not exist, or when it exists, it may not equal to the value of the function f(x) at that point. Necessary and sufficient conditions for a function to be represented by a Fourier series are still waiting to be discovered. One sufficient condition (not the most general one) is presented in the following statement. Theorem 5.. Suppose that a periodic function f with a period is a piecewise continuous function as is its first derivative f. Then the function f is represented by a convergent series on the interval [, ], the Fourier series (5..), whose coefficients are given by a n = b n = f(x) cos nπx f(x) sin nπx dx, n =,,,...; (5..) dx, n =,,... (5..) The Fourier series converges to f(x) at all points f is continuous and to at all points f is discontinuous. [f(x+)+f(x )] = lim ǫ [f(x+ǫ)+f(x ǫ)]
2 APMA Week #5 Spring For uniform convergence of the Fourier series, we need to impose an additional condition on the function. Theorem 5.. et f be a continuous function on (, ) and periodic with a period of. If f is a piecewise continuous function on [,], then the Fourier series (5..) converges uniformly to f on [,] and hence on any interval. That is, for every ε >, there exists an integer N = N (ε) that depends on ε such that the partial sums S N (x) = for all N N, and all x (, ). a + N n= ( +b n sin nπx ) < ε, The series (5..) thereby defines a periodic function, but it may not be differentiable, or even continuous. Recall that a function on an interval [, ] is piecewise continuous if the interval can be partitioned by a finite number of subintervals so that on each of it f is continuous, and approaches finite limits at the end points. So tanx is not a piecewise continuous function because it has infinite jumps at points kπ +π/, k =,±,±,... The American mathematician Josiah Willard Gibbs (89 9) observed in 898 that near points of discontinuity of f(x), the Nth partial sums S N (x) of the Fourier series for f may overshoot/undershoot by approximately 9% of the jump, regardless of the number of terms, N. This is the Gibbs phenomenon, which was first noticed and analyzed by an English mathematician Henry Wilbraham (85 88) in Even and Odd Functions Recall that f is called an even function if its domain contains the point x whenever it contains the point x, and if f( x) = f(x) for each x in the domain of f. Similarly, f is said to be an odd function if its domain contains the point x whenever it contains the point x, and if f( x) = f(x) for each x in the domain of f. Any function that is a linear combination of monomials x p with even (odd) powers p is an even (odd) function. Since Taylor s series for cosine function cosx = k ( ) k xk (k)! contains only even powers, it is an even function. Similarly, the sine function sinx = ( ) k x k+ is an example of (k +)! k an odd function. A sum or difference of two or more even functions is an even function; for instance, x + cosx is an even function. On the other hand, a product of two even or odd functions is an even function, while a product of an even and an odd function is an odd function. For instance, sin x and cos x are even functions, while sinx cosx is an odd function.
3 APMA Week #5 Spring When an odd function is represented by a Fourier series, its expansion will be a sine Fourier series, so all coefficients of cosine terms are zeroes: a = and a n = for all n; hence, f(x) = n b k sin kπx, (5..) b k = f(x) sin kπx dx = f(x) sin kπx dx, k =,,... (5..) We refer to this series (5..) as a Fourier sine series. It can be considered as a series for the function f(x) with a domain of the interval [,], and extended in odd manner to the interval [,] (that is, f( x) = f(x)). Similarly, if a function g(x) is an even function on an interval [,], then its Fourier series contains only cosine functions. Therefore such series is called Fourier cosine series (all coefficients b n in Eq. (5..) are zeroes): a n = g(x) = a + a n sin nπx, (5..) n f(x) cos nπx dx, n =,,,... (5..) The same function can be extended in either odd way or even way on the interval [,]; so the same function may have different Fourier series representation. Example 5.. Consider the function f(x) = x on the interval [,]. First, we extend it in an even way (see Fig. (a)), which leads to the Fourier cosine series: a = a n = x = a +, n x dx = 8, x cos nπx This yields the following cosine series: x = + 6 π For N >, its partial sum approximations are 6 6 dx = cos(nπ) = n π n π ( )n. n ( ) n n cos nπx. x C N (x) = + 6 π N ( ) n n= n cos nπx. (5..5)
4 APMA Week #5 Spring (a) (b) (c) Figure : Extensions of the function x to the negative semiaxis: (a) even (b) odd, and (c) periodic way. Figure : Partial Fourier cosine sum approximations (5..5) for N = (black) and N = (blue) of the function f = x : 5 Now we extend the function x into negative semiaxis in odd way (see Fig. (b)), which leads to the sine Fourier series: x = b n sin nπx, n Therefore, b n = x sin nπx x = 8 ( ) n π n n dx = 6 n π [( )n ] 8 nπ ( )n. sin nπx (k +)πx sin π (k +) and its Nth partial sums become (after using only odd indices: n = k + in the latter sum) x S N (x) = 8 π N ( ) n n= n k sin nπx (N )/ (k +)πx sin π (k +) k= For the periodic extension (see Fig (c)) with half period =, we have the general Fourier series: x = A + [A n cos(nπx)+b n sin(nπx)], n,.
5 APMA Week #5 Spring This leads to A = A n = B n = (x+) dx+ x dx = (x+) cos(nπx)dx+ (x+) sin(nπx)dx+ x + π N n= n cos(nπx) π x dx = 8, x cos(nπx)dx = n π, x sin(nπx)dx = nπ. N n= n sin(nπx). To estimate the quality of such approximations, let us calculate the partial sums for different values of N at x = : At the point x = ( the given function x is continuous)), we have The Fourier series partial sums (except cosine series because even extension of x is a continuous function) demonstrate the Gibbs phenomenon near the points of discontinuity x = and x =. For instance, at x =.9, we have
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