12.2 Fourier Series

INTRODUCTION

We have just seen in the preceding section that if {ø0(x), ø1(x), ø2(x), …} is a set of real-valued functions that is orthogonal on an interval [a, b] and if f is a function defined on the same interval, then we can formally expand f in an orthogonal series c0ø0(x) + c1ø1(x) + c2ø2(x) + … . In this section we shall expand functions in terms of a special orthogonal set of trigonometric functions.

Trigonometric Series

In Problem 12 in Exercises 12.1, you were asked to show that the set of trigonometric functions

(1)

is orthogonal on the interval [–p, p]. This set will be of special importance later on in the solution of certain kinds of boundary-value problems involving linear partial differential equations. In those applications we will need to expand a function f defined on [–p, p] in an orthogonal series consisting of the trigonometric functions in (1); that is,

(2)

The coefficients a0, a1, a2, …, b1, b2, …, can be determined in exactly the same manner as in the general discussion of orthogonal series expansions on pages 683 and 684. Before proceeding, note that we have chosen to write the coefficient of 1 in the set (1) as rather than a0; this is for convenience only because the formula of an will then reduce to a0 for n = 0.

This is why a0 is used instead of a0.

Now integrating both sides of (2) from –p to p gives

(3)

Since cos(nπx/p) and sin(nπx/p), n ≥ 1, are orthogonal to 1 on the interval, the right side of (3) reduces to a single term:

Solving for a0 yields

(4)

Now we multiply (2) by cos(mπx/p) and integrate:

(5)

By orthogonality we have

and

Thus (5) reduces to

and so (6)

Finally, if we multiply (2) by sin(mπx/p), integrate, and make use of the results

and

we find that (7)

The trigonometric series (2) with coefficients defined by (4), (6), and (7), respectively, is said to be the Fourier series of the function f. The coefficients obtained from (4), (6), and (7) are referred to as Fourier coefficients of f. If the Fourier series (2) converges, it is sometimes called a representation of f.

Although the French mathematical physicist and scientific advisor to Napoleon Bonaparte, Jean-Baptiste Joseph Fourier (1768–1830), did not invent the series that bears his name, he was at least responsible for sparking the interest of mathematicians in trigonometric series by his less-than-rigorous use of them in his research on the conduction of heat.

In finding the coefficients a0, an, and bn , we assumed that f was integrable on the interval and that (2), as well as the series obtained by multiplying (2) by cos (mπx/p), converged in such a manner as to permit term-by-term integration. Until (2) is shown to be convergent for a given function f, the equality sign is not to be taken in a strict or literal sense. Some texts use the symbol ~ in place of =. In view of the fact that most functions in applications are of a type that guarantees convergence of the series, we shall use the equality symbol. We summarize the results:

DEFINITION 12.2.1 Fourier Series

The Fourier series of a function f defined on the interval (–p, p) is given by

(8)

where (9)

(10)

(11)

EXAMPLE 1 Expansion in a Fourier Series

Expand (12)

in a Fourier series.

SOLUTION

The graph of f is given in FIGURE 12.2.1. With p = π we have from (9) and (10) that

A function f is graphed on an x y plane. Two parts of the function are shown as two straight lines. The first line is horizontal and starts at the point (negative pi, 0) and ends at the point (0, 0). The second line starts at the point (0, pi), goes down and to the right, and ends at the point (pi, 0).

FIGURE 12.2.1 Function f in Example 1

In like manner we find from (11) that

Note that 1 –(–1)n

Therefore (13)≡

Note that an defined by (10) reduces to a0 given by (9) when we set n = 0. But as Example 1 shows, this may not be the case after the integral for an is evaluated.

Convergence of a Fourier Series

The following theorem gives sufficient conditions for convergence of a Fourier series at a point.

THEOREM 12.2.1 Conditions for Convergence

Let f and f′ be piecewise continuous on the interval [–p, p]; that is, let f and f′ be continuous except at a finite number of points in the interval and have only finite discontinuities at these points. Then for all x in the interval (–p, p) the Fourier series of f converges to f(x) at a point of continuity. At a point of discontinuity, the Fourier series converges to the average

where f(x+) and f(x–) denote the limit of f at x from the right and from the left, respectively.*

For a proof of this theorem you are referred to the classic text by Churchill and Brown.†

EXAMPLE 2 Convergence of a Point of Discontinuity

The function (12) in Example 1 satisfies the conditions of Theorem 12.2.1. Thus for every x in the interval (–π, π), except at x = 0, the series (13) will converge to f(x). At x = 0 the function is discontinuous, and so the series (13) will converge to

≡

Periodic Extension

Observe that each of the functions in the basic set (1) has a different fundamental period,* namely, 2p/n, n ≥ 1, but since a positive integer multiple of a period is also a period we see that all of the functions have in common the period 2p (verify). Hence the right-hand side of (2) is 2p-periodic; indeed, 2p is the fundamental period of the sum. We conclude that a Fourier series not only represents the function on the interval (–p, p) but also gives the periodic extension of f outside this interval. We can now apply Theorem 12.2.1 to the periodic extension of f, or we may assume from the outset that the given function is periodic with period T = 2p; that is, f(x + T) = f(x). When f is piecewise continuous and the right- and left-hand derivatives exist at x = –p and x = p, respectively, then the series (8) converges to the average [f(p–) + f(–p+)]/2 at these endpoints and to this value extended periodically to ±3p, ±5p, ±7p, and so on. The Fourier series in (13) converges to the periodic extension of (12) on the entire x-axis. At 0, ±2π, ±4π, …, and at ±π, ±3π, ±5π, …, the series converges to the values

We may assume that the given function f is periodic.

respectively. The solid dots in FIGURE 12.2.2 represent the value π/2.

Five parts of a function f are graphed on an x y plane. The first part represented by dashed red line starts at the point (negative 4 pi, pi), goes down and to the right, and reaches the point (negative 3 pi, 0). Then it goes to the right, and ends at (negative 2 pi, 0). The second part starts at the point (negative 2 pi, pi), goes down and to the right, and reaches the point (negative pi, 0). Then it goes to the right, and ends at (0, 0). From (negative 2 pi, pi) to (negative pi, 0), it is represented by a dashed red line. From (negative pi, 0) to (0, 0), it is represented by a solid blue line. The third part starts at the point (0, pi), goes down and to the right, and reaches the point (pi, 0). Then it goes to the right, and ends at (2 pi, 0). From (0, pi) to (pi, 0), it is represented by a solid blue line. From (pi, 0) to (2 pi, 0), it is represented by a dashed red line. The fourth part represented by dashed red line starts at the point (2 pi, pi), goes down and to the right, and reaches the point (3 pi, 0). Then it goes to the right, and ends at (4 pi, 0). The fifth part represented by a dashed red line starts at the point (4 pi, pi), goes down and to the right, and ends at the point (5 pi, 0). Five solid dots are placed at the following points: (negative 4 pi, pi over 2), (negative 2 pi, pi over 2), (0, pi over 2), (2 pi, pi over 2), and (4 pi, pi over 2).

FIGURE 12.2.2 Periodic extension of the function f shown in Figure 12.2.1

Sequence of Partial Sums

It is interesting to see how the sequence of partial sums {SN(x)} of a Fourier series approximates a function. For example, the first three partial sums of (13) are

S1(x) = , S2(x) = + cos x + sin x, S3(x) = + cos x + sin x + sin 2x.

In FIGURE 12.2.3 we have used a CAS to graph the partial sums S5(x), S8(x), and S15(x) of (13) on the interval (–π, π). Figure 12.2.3(d) shows the periodic extension using S15(x) on (–4π, 4π).

The first graph is captioned (a) S subscript 5 (x) on (negative pi, pi). A curve is graphed on an x y plane. It starts approximately at (negative 3, 0.1), follows an increasing oscillatory pattern, and reaches a high point in the first quadrant through the positive y axis. Then it goes down and to the right, and ends approximately at (3, 0.1). The second graph is captioned (b) S subscript 8 (x) on (negative pi, pi). A curve is graphed on an x y plane. It starts approximately at (negative 3, 0.1), follows an increasing oscillatory pattern, and reaches a high point in the first quadrant through the positive y axis. Then it goes down and to the right with slight oscillations, and ends approximately at (3, 0.1). The third graph is captioned (c) S subscript 15 (x) on (negative pi, pi). A curve is graphed on an x y plane. It starts approximately at (negative 3, 0), follows an increasing oscillatory pattern, and reaches a high point in the first quadrant through the positive y axis. Then it goes down and to the right with slight oscillations, and ends approximately at (3, 0). The fourth graph is captioned (d) S subscript 15 (x) on (negative 4pi, 4pi). A curve is graphed on an x y plane. It starts approximately at (negative 13, 1.5), goes up and to the right, and goes down and to the right with slight oscillations to the approximate point (negative 10, 0). Then it follows an increasing oscillatory pattern, and reaches a high point in the second quadrant. This pattern is repeated four times and the curve ends approximately at (13, 1.5). For the increasing oscillatory patterns in the four curves, the length of oscillations is longer in the first graph but the length becomes shorter in the successive second and third graphs, and is the shortest in the fourth graph.

FIGURE 12.2.3 Partial sums of a Fourier series

12.2 Exercises Answers to selected odd-numbered problems begin on page ANS-31.

In Problems 1–18, find the Fourier series of the function f on the given interval. Give the number to which the Fourier series converges at a point of discontinuity of f.

In Problems 19 and 20, sketch the periodic extension of the indicated function.

  1. The function f in Problem 9
  2. The function f in Problem 14
  3. Use the result of Problem 5 to show

    and

  4. Use Problem 21 to find a series that gives the numerical value of π2/8.
  5. Use the result of Problem 7 to show

  6. Use the result of Problem 9 to show

    .

  7. The root–mean–square value of a function f(x) defined over an interval (a, b) is given by

    If the Fourier series expansion of f is given by (8), show that the RMS value of f over the interval (–p, p) is given by

    where a0, an , and bn are the Fourier coefficients in (9), (10), and (11), respectively.

 

*In other words, for x a point in the interval and h > 0,

f(x+) = f(x + h), f(x–) = f(x – h).

†Ruel V. Churchill and James Ward Brown, Fourier Series and Boundary Value Problems (New York: McGraw-Hill, 2011).

*See Problems 21–26 in Exercises 12.1.