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To guarantee convergence of a Fourier series to the function from which its coefficients were computed it is essential to place additional conditions on the function. From a practical point of view, such conditions should be broad enough to cover all situations of interest, yet simple enough to be easily checked for particular functions. Through the years several sets of conditions have been devised to serve this purpose.
Before stating a convergence theorem for Fourier series, we define a term that appears in the theorem. A function f is said to be piecewise continuous on an interval a < x < b if the interval can be partitioned by a finite number of points a = x0 < x1 < < xn = b so that
1. f is continuous on each open subinterval xi 1 < x < xi .
2. f approaches a finite limit as the endpoints of each subinterval are approached from within the subinterval.
The graph of a piecewise continuous function is shown in Figure 10.3.1.
10.3 The Fourier Convergence Theorem
The notation f (c+) is used to denote the limit of f (x) as x ^ c from the right; similarly, f (c-) denotes the limit of f (x) as x approaches c from the left.
Note that it is not essential that the function even be defined at the partition points xt. For example, in the following theorem we assume that f' is piecewise continuous; but certainly f' does not exist at those points where f itself is discontinuous. It is also not essential that the interval be closed; it may also be open, or open at one end and closed at the other.
Theorem 10.3.1 Suppose that f and f' are piecewise continuous on the interval L < x < L .Further,
suppose that f is defined outside the interval L < x < L so that it is periodic with
period 2L. Then f has a Fourier series
/?/x ao ^ / mnx . mnx \
f (x) = 2° + ^ \m cos ~T~ + bm sin ~^) , (4)
m = 1
whose coefficients are given by Eqs. (2) and (3). The Fourier series converges to f (x) at all points where f is continuous, and to [f (x+) + f (x)]/2 at all points where f is discontinuous.
Note that [ f (x+) + f (x)]/2 is the mean value of the right- and left-hand limits at the point x. At any point where f is continuous, f (x+) = f (x) = f (x). Thus it is correct to say that the Fourier series converges to [ f (x+) + f (x)]/2 at all points. Whenever we say that a Fourier series converges to a function f, we always mean that it converges in this sense.
It should be emphasized that the conditions given in this theorem are only sufficient for the convergence of a Fourier series; they are by no means necessary. Neither are they the most general sufficient conditions that have been discovered. In spite of this, the proof of the theorem is fairly difficult and is not given here.3
To obtain a better understanding of the content of the theorem it is helpful to consider some classes of functions that fail to satisfy the assumed conditions. Functions that are not included in the theorem are primarily those with infinite discontinuities in the interval [L, L], such as 1/x2 as x ^ 0, or ln |x L | as x ^ L. Functions having an infinite number of jump discontinuities in this interval are also excluded; however, such functions are rarely encountered.
3Proofs of the convergence of a Fourier series can be found in most books on advanced calculus. See, for example, Kaplan (Chapter 7) or Buck (Chapter 6).
Chapter 10. Partial Differential Equations and Fourier Series
It is noteworthy that a Fourier series may converge to a sum that is not differentiable, or even continuous, in spite of the fact that each term in the series (4) is continuous, and even differentiable infinitely many times. The example below is an illustration of this, as is Example 2 in Section 10.2.
f (x ) =
0, L < x < 0,
L, 0 < x < L,
and let f be defined outside this interval so that f (x + 2L) = f (x) for all x. We will temporarily leave open the definition of f at the points x = 0, ħL, except that its value must be finite. Find the Fourier series for this function and determine where it converges.
1 1 1 III ,
1 CXI 1 CO 1 L 2L 3L x
FIGURE 10.3.2 Square wave.
The equation y = f (x) has the graph shown in Figure 10.3.2, extended to infinity in both directions. It can be thought of as representing a square wave. The interval [L, L] can be partitioned to give the two open subintervals (-L, 0) and (0, L). In (0, L ), f (x) = L and f(x) = 0. Clearly, both f and f' are continuous and furthermore have limits as x ^ 0 from the right and as x ^ L from the left. The situation in (-L, 0) is similar. Consequently, both f and f are piecewise continuous on [L, L), so f satisfies the conditions of Theorem 10.3.1. If the coefficients am and bm are computed from Eqs. (2) and (3), the convergence of the resulting Fourier series to f (x ) is assured at all points where f is continuous. Note that the values of am and bm are the same regardless of the definition of f at its points of discontinuity. This is true because the value of an integral is unaffected by changing the value of the integrand at a finite number of points. From Eq. (2)