CHAPTER 11

Fourier Analysis

This chapter on Fourier analysis covers three broad areas: Fourier series in Secs. 11.1–11.4, more general orthonormal series called Sturm–Liouville expansions in Secs. 11.5 and 11.6 and Fourier integrals and transforms in Secs. 11.7–11.9.

The central starting point of Fourier analysis is Fourier series. They are infinite series designed to represent general periodic functions in terms of simple ones, namely, cosines and sines. This trigonometric system is orthogonal, allowing the computation of the coefficients of the Fourier series by use of the well-known Euler formulas, as shown in Sec. 11.1. Fourier series are very important to the engineer and physicist because they allow the solution of ODEs in connection with forced oscillations (Sec. 11.3) and the approximation of periodic functions (Sec. 11.4). Moreover, applications of Fourier analysis to PDEs are given in Chap. 12. Fourier series are, in a certain sense, more universal than the familiar Taylor series in calculus because many discontinuous periodic functions that come up in applications can be developed in Fourier series but do not have Taylor series expansions.

The underlying idea of the Fourier series can be extended in two important ways. We can replace the trigonometric system by other families of orthogonal functions, e.g., Bessel functions and obtain the Sturm–Liouville expansions. Note that related Secs. 11.5 and 11.6 used to be part of Chap. 5 but, for greater readability and logical coherence, are now part of Chap. 11. The second expansion is applying Fourier series to nonperiodic phenomena and obtaining Fourier integrals and Fourier transforms. Both extensions have important applications to solving PDEs as will be shown in Chap. 12.

In a digital age, the discrete Fourier transform plays an important role. Signals, such as voice or music, are sampled and analyzed for frequencies. An important algorithm, in this context, is the fast Fourier transform. This is discussed in Sec. 11.9.

Note that the two extensions of Fourier series are independent of each other and may be studied in the order suggested in this chapter or by studying Fourier integrals and transforms first and then Sturm–Liouville expansions.

Prerequisite: Elementary integral calculus (needed for Fourier coefficients).

Sections that may be omitted in a shorter course: 11.4–11.9.

References and Answers to Problems: App. 1 Part C, App. 2.

11.1 Fourier Series

Fourier series are infinite series that represent periodic functions in terms of cosines and sines. As such, Fourier series are of greatest importance to the engineer and applied mathematician. To define Fourier series, we first need some background material. A function f(x) is called a periodic function if f(x) is defined for all real x, except

Fig. 258. Periodic function of period p

possibly at some points, and if there is some positive number p, called a period of, f(x) such that

(The function f(x) = tan x is a periodic function that is not defined for all real x but undefined for some points (more precisely, countably many points), that is x = ±π/2, ±π/2, ….)

The graph of a periodic function has the characteristic that it can be obtained by periodic repetition of its graph in any interval of length p (Fig. 258).

The smallest positive period is often called the fundamental period. (See Probs. 2–4.)

Familiar periodic functions are the cosine, sine, tangent, and cotangent. Examples of functions that are not periodic are x, x², x³, e^x, cosh x, and In x, to mention just a few.

If f(x) has period p, it also has the period 2p because (1) implies f(x + 2p) = f([x + p] + p) = f(x + p) = f(x), etc.; thus for any integer n = 1, 2, 3, …,

Furthermore if f(x) and g(x) have period p, then af(x) + bg(x) with any constants a and b also has the period p.

Our problem in the first few sections of this chapter will be the representation of various functions f(x) of period 2π in terms of the simple functions

All these functions have the period 2π. They form the so-called trigonometric system. Figure 259 shows the first few of them (except for the constant 1, which is periodic with any period).

Fig. 259. Cosine and sine functions having the period 2π (the first few members of the trigonometric system (3), except for the constant 1)

The series to be obtained will be a trigonometric series, that is, a series of the form

a₀, a₁, b₁, a₂, b₂, … are constants, called the coefficients of the series. We see that each term has the period 2π. Hence if the coefficients are such that the series converges, its sum will be a function of period 2π.

Expressions such as (4) will occur frequently in Fourier analysis. To compare the expression on the right with that on the left, simply write the terms in the summation. Convergence of one side implies convergence of the other and the sums will be the same.

Now suppose that f(x) is a given function of period 2π and is such that it can be represented by a series (4), that is, (4) converges and, moreover, has the sum f(x). Then, using the equality sign, we write

and call (5) the Fourier series of f(x). We shall prove that in this case the coefficients of (5) are the so-called Fourier coefficients of f(x), given by the Euler formulas

The name “Fourier series” is sometimes also used in the exceptional case that (5) with coefficients (6) does not converge or does not have the sum f(x)—this may happen but is merely of theoretical interest. (For Euler see footnote 4 in Sec. 2.5.)

A Basic Example

Before we derive the Euler formulas (6), let us consider how (5) and (6) are applied in this important basic example. Be fully alert, as the way we approach and solve this example will be the technique you will use for other functions. Note that the integration is a little bit different from what you are familiar with in calculus because of the n. Do not just routinely use your software but try to get a good understanding and make observations: How are continuous functions (cosines and sines) able to represent a given discontinuous function? How does the quality of the approximation increase if you take more and more terms of the series? Why are the approximating functions, called the partial sums of the series, in this example always zero at 0 and π? Why is the factor 1/n (obtained in the integration) important?

Fig. 261. First three partial sums of the corresponding Fourier series

Derivation of the Euler Formulas (6)

The key to the Euler formulas (6) is the orthogonality of (3), a concept of basic importance, as follows. Here we generalize the concept of inner product (Sec. 9.3) to functions.

Application of Theorem 1 to the Fourier Series (5)

We prove (6.0). Integrating on both sides of (5) from to −π to π, we get

We now assume that termwise integration is allowed. (We shall say in the proof of Theorem 2 when this is true.) Then we obtain

The first term on the right equals 2πa₀. Integration shows that all the other integrals are 0. Hence division by 2π gives (6.0).

We prove (6a). Multiplying (5) on both sides by cos mx with any fixed positive integer m and integrating from −π to π, we have

We now integrate term by term. Then on the right we obtain an integral of a₀ cos mx which is 0; an integral of a_n cos nx cos mx, which is a_mπ for n = m and 0 for n ≠ m by (9a); and an integral of b_n sin nx cos mx, which is 0 for all n and m by (9c). Hence the right side of (10) equals a_mπ. Division by π gives (6a) (with m instead of n).

We finally prove (6b). Multiplying (5) on both sides by sin mx with any fixed positive integer m and integrating from −π to π, we get

Integrating term by term, we obtain on the right an integral of a₀ sin mx, which is 0; an integral of a_n cos nx sin mx, which is 0 by (9c); and an integral of b_n sin nx sin mx, which is b_mπ if n = m and 0 if n ≠ m, by (9b). This implies (6b) (with n denoted by m). This completes the proof of the Euler formulas (6) for the Fourier coefficients.

Convergence and Sum of a Fourier Series

The class of functions that can be represented by Fourier series is surprisingly large and general. Sufficient conditions valid in most applications are as follows.

Fig. 262. Left- and right-hand limits of the function

Summary. A Fourier series of a given function f(x) of period 2π is a series of the form (5) with coefficients given by the Euler formulas (6). Theorem 2 gives conditions that are sufficient for this series to converge and at each x to have the value f(x), except at discontinuities of f(x), where the series equals the arithmetic mean of the left-hand and right-hand limits of f(x) at that point.

11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions

We now expand our initial basic discussion of Fourier series.

Orientation. This section concerns three topics:

1. Transition from period 2π to any period 2L, for the function f, simply by a transformation of scale on the x-axis.
2. Simplifications. Only cosine terms if f is even (“Fourier cosine series”). Only sine terms if f is odd (“Fourier sine series”).
3. Expansion of f given for 0 x L in two Fourier series, one having only cosine terms and the other only sine terms (“half-range expansions”).

1. From Period 2π to Any Period p = 2L

Clearly, periodic functions in applications may have any period, not just 2π as in the last section (chosen to have simple formulas). The notation p = 2L for the period is practical because L will be a length of a violin string in Sec. 12.2, of a rod in heat conduction in Sec. 12.5, and so on.

The transition from period 2π to be period p = 2L is effected by a suitable change of scale, as follows. Let f(x) have period p = 2L. Then we can introduce a new variable υ such that f(x), as a function of υ, has period 2π. If we set

then υ = ±π corresponds to x = ±L. This means that f, as a function of υ, has period 2π and, therefore, a Fourier series of the form

with coefficients obtained from (6) in the last section

We could use these formulas directly, but the change to x simplifies calculations. Since

and we integrate over x from −L to L. Consequently, we obtain for a function f(x) of period 2L the Fourier series

with the Fourier coefficients of f(x) given by the Euler formulas (π/L in dx cancels 1/π in (3))

Just as in Sec. 11.1, we continue to call (5) with any coefficients a trigonometric series. And we can integrate from 0 to 2L or over any other interval of length p = 2L.

Fig. 263. Example 1

Fig. 264. Example 2

Fig. 265. Half-wave rectifier

2. Simplifications: Even and Odd Functions

If f(x) is an even function, that is, f(−x) = f(x) (see Fig. 266), its Fourier series (5) reduces to a Fourier cosine series

with coefficients (note: integration from 0 to L only!)

Fig. 266. Even function

Fig. 267. Odd function

If f(x) is an odd function, that is, f(−x) = −f(x) (see Fig. 267), its Fourier series (5) reduces to a Fourier sine series

with coefficients

These formulas follow from (5) and (6) by remembering from calculus that the definite integral gives the net area (= area above the axis minus area below the axis) under the curve of a function between the limits of integration. This implies

Formula (7b) implies the reduction to the cosine series (even f makes f(x) sin (nπx/L) odd since sin is odd) and to the sine series (odd f makes f(x) cos (nπx/L) odd since cos is even.) Similarly, (7a) reduces the integrals in (6*) and (6**) to integrals from 0 to L. These reductions are obvious from the graphs of an even and an odd function. (Give a formal proof.)

Further simplifications result from the following property, whose very simple proof is left to the student.

3. Half-Range Expansions

Half-range expansions are Fourier series. The idea is simple and useful. Figure 270 explains it. We want to represent f(x) in Fig. 270. 0 by a Fourier series, where f(x) may be the shape of a distorted violin string or the temperature in a metal bar of length L, for example. (Corresponding problems will be discussed in Chap. 12.) Now comes the idea.

We could extend f(x) as a function of period L and develop the extended function into a Fourier series. But this series would, in general, contain both cosine and sine terms. We can do better and get simpler series. Indeed, for our given f we can calculate Fourier coefficients from (6*) or from (6**). And we have a choice and can take what seems more practical. If we use (6*), we get (5*). This is the even periodic extension f₁ of f in Fig. 270a. If we choose (6**) instead, we get the (5**), odd periodic extension f₂ of f in Fig. 270b.

Both extensions have period 2L. This motivates the name half-range expansions: f is given (and of physical interest) only on half the range, that is, on half the interval of periodicity of length 2L.

Let us illustrate these ideas with an example that we shall also need in Chap. 12.

Fig. 270. Even and odd extensions of period 2L

Fig. 272. Periodic extensions of f(x) in Example 6

11.3 Forced Oscillations

Fourier series have important applications for both ODEs and PDEs. In this section we shall focus on ODEs and cover similar applications for PDEs in Chap. 12. All these applications will show our indebtedness to Euler's and Fourier's ingenious idea of splitting up periodic functions into the simplest ones possible.

From Sec. 2.8 we know that forced oscillations of a body of mass m on a spring of modulus k are governed by the ODE

where y = y(t) is the displacement from rest, c the damping constant, k the spring constant (spring modulus), and r(t) the external force depending on time t. Figure 274 shows the model and Fig. 275 its electrical analog, an RLC- circuit governed by

We consider (1). If r(t) is a sine or cosine function and if there is damping (c > 0), then the steady-state solution is a harmonic oscillation with frequency equal to that of r(t). However, if r(t) is not a pure sine or cosine function but is any other periodic function, then the steady-state solution will be a superposition of harmonic oscillations with frequencies equal to that of r(t) and integer multiples of these frequencies. And if one of these frequencies is close to the (practical) resonant frequency of the vibrating system (see Sec. 2.8), then the corresponding oscillation may be the dominant part of the response of the system to the external force. This is what the use of Fourier series will show us. Of course, this is quite surprising to an observer unfamiliar with Fourier series, which are highly important in the study of vibrating systems and resonance. Let us discuss the entire situation in terms of a typical example.

Fig. 274. Vibrating system under consideration

Fig. 275. Electrical analog of the system in Fig. 274 (RLC-circuit)

Fig. 277. Input and steady-state output in Example 1

11.4 Approximation by Trigonometric Polynomials

Fourier series play a prominent role not only in differential equations but also in approximation theory, an area that is concerned with approximating functions by other functions—usually simpler functions. Here is how Fourier series come into the picture.

Let f(x) be a function on the interval −π x π that can be represented on this interval by a Fourier series. Then the Nth partial sum of the Fourier series

is an approximation of the given f(x). In (1) we choose an arbitrary N and keep it fixed. Then we ask whether (1) is the “best” approximation of f by a trigonometric polynomial of the same degree N, that is, by a function of the form

Here, “best” means that the “error” of the approximation is as small as possible.

Of course we must first define what we mean by the error of such an approximation. We could choose the maximum of |f(x) − F(x)|. But in connection with Fourier series it is better to choose a definition of error that measures the goodness of agreement between f and F on the whole interval −π x π. This is preferable since the sum f of a Fourier series may have jumps: F in Fig. 278 is a good overall approximation of f, but the maximum of |f(x) − F(x)| (more precisely, the supremum) is large. We choose

Fig. 278. Error of approximation

This is called the square error of F relative to the function f on the interval −π x π. Clearly, E 0.

N being fixed, we want to determine the coefficients in (2) such that E is minimum. Since (f − F)² = f² − 2fF + F², we have

We square (2), insert it into the last integral in (4), and evaluate the occurring integrals. This gives integrals of cos² nx and sin² nx(n 1), which equal π, and integrals of cos nx, sin nx, and (cos nx)(sin mx), which are zero (just as in Sec. 11.1). Thus

We now insert (2) into the integral of f F in (4). This gives integrals of f cos nx as well as f sin nx, just as in Euler's formulas, Sec. 11.1, for a_n and b_n (each multiplied by A_n or B_n).

With these expressions, (4) becomes

We now take A_n = a_n and B_n = b_n in (2). Then in (5) the second line cancels half of the integral-free expression in the first line. Hence for this choice of the coefficients of F the square error, call it E*, is

We finally subtract (6) from (5). Then the integrals drop out and we get terms and similar terms (B_n − b_n)²:

Since the sum of squares of real numbers on the right cannot be negative,

and E = E* if and only if A₀ = a₀, …, B_N = b_N. This proves the following fundamental minimum property of the partial sums of Fourier series.

From (6) we see that E* cannot increase as N increases, but may decrease. Hence with increasing N the partial sums of the Fourier series of f yield better and better approximations to f, considered from the viewpoint of the square error.

Since E* 0 and (6) holds for every N, we obtain from (6) the important Bessel's inequality

for the Fourier coefficients of any function f for which integral on the right exists. (For F. W. Bessel see Sec. 5.5.)

It can be shown (see [C12] in App. 1) that for such a function f, Parseval's theorem holds; that is, formula (7) holds with the equality sign, so that it becomes Parseval's identity³

11.5 Sturm–Liouville Problems. Orthogonal Functions

The idea of the Fourier series was to represent general periodic functions in terms of cosines and sines. The latter formed a trigonometric system. This trigonometric system has the desirable property of orthogonality which allows us to compute the coefficient of the Fourier series by the Euler formulas.

The question then arises, can this approach be generalized? That is, can we replace the trigonometric system of Sec. 11.1 by other orthogonal systems (sets of other orthogonal functions)? The answer is “yes” and will lead to generalized Fourier series, including the Fourier–Legendre series and the Fourier–Bessel series in Sec. 11.6.

To prepare for this generalization, we first have to introduce the concept of a Sturm–Liouville Problem. (The motivation for this approach will become clear as you read on.) Consider a second-order ODE of the form

on some interval a x b, satisfying conditions of the form

Here λ is a parameter, and k₁, k₂, l₁, l₂ are given real constants. Furthermore, at least one of each constant in each condition (2) must be different from zero. (We will see in Example 1 that, if p(x) = r(x) = 1 and q(x) = 0, then sin and cos satisfy (1) and constants can be found to satisfy (2).) Equation (1) is known as a Sturm–Liouville equation.⁴ Together with conditions 2(a), 2(b) it is know as the Sturm–Liouville problem. It is an example of a boundary value problem.

A boundary value problem consists of an ODE and given boundary conditions referring to the two boundary points (endpoints) x = a and x = b of a given interval a x b.

The goal is to solve these type of problems. To do so, we have to consider

Eigenvalues, Eigenfunctions

Clearly, y ≡ 0 is a solution—the “trivial solution”—of the problem (1), (2) for any λ because (1) is homogeneous and (2) has zeros on the right. This is of no interest. We want to find eigenfunctions y(x), that is, solutions of (1) satisfying (2) without being identically zero. We call a number λ for which an eigenfunction exists an eigenvalue of the Sturm–Liouville problem (1), (2).

Many important ODEs in engineering can be written as Sturm–Liouville equations. The following example serves as a case in point.

Note that the solution to this problem is precisely the trigonometric system of the Fourier series considered earlier. It can be shown that, under rather general conditions on the functions p, q, r in (1), the Sturm–Liouville problem (1), (2) has infinitely many eigenvalues. The corresponding rather complicated theory can be found in Ref. [All] listed in App. 1.

Furthermore, if p, q, r, and p′ in (1) are real-valued and continuous on the interval a x b and r is positive throughout that interval (or negative throughout that interval), then all the eigenvalues of the Sturm–Liouville problem (1), (2) are real. (Proof in App. 4.) This is what the engineer would expect since eigenvalues are often related to frequencies, energies, or other physical quantities that must be real.

The most remarkable and important property of eigenfunctions of Sturm–Liouville problems is their orthogonality, which will be crucial in series developments in terms of eigenfunctions, as we shall see in the next section. This suggests that we should next consider orthogonal functions.

Orthogonal Functions

Functions y₁(x), y₂(x), … defined on some interval a x b are called orthogonal on this interval with respect to the weight function r(x) > 0 if for all m and all n different from m,

(y_m, y_n) is a standard notation for this integral. The norm ||y_m|| of y_m is defined by

Note that this is the square root of the integral in (4) with n = m.

The functions y₁, y₂, … are called orthonormal on a x b if they are orthogonal on this interval and all have norm 1. Then we can write (4), (5) jointly by using the Kronecker symbol⁵ δ_mn, namely,

If r(x) = 1, we more briefly call the functions orthogonal instead of orthogonal with respect to r(x) = 1; similarly for orthognormality. Then

The next example serves as an illustration of the material on orthogonal functions just discussed.

Theorem 1 shows that for any Sturm–Liouville problem, the eigenfunctions associated with these problems are orthogonal. This means, in practice, if we can formulate a problem as a Sturm–Liouville problem, then by this theorem we are guaranteed orthogonality.

The boundary value problem consisting of the Sturm–Liouville equation (1) and the periodic boundary conditions (7) is called a periodic Sturm–Liouville problem.

Example 3 confirms, from this new perspective, that the trigonometric system underlying the Fourier series is orthogonal, as we knew from Sec. 11.1.

What we have seen is that the trigonometric system, underlying the Fourier series, is a solution to a Sturm–Liouville problem, as shown in Example 1, and that this trigonometric system is orthogonal, which we knew from Sec. 11.1 and confirmed in Example 3.

11.6 Orthogonal Series. Generalized Fourier Series

Fourier series are made up of the trigonometric system (Sec. 11.1), which is orthogonal, and orthogonality was essential in obtaining the Euler formulas for the Fourier coefficients. Orthogonality will also give us coefficient formulas for the desired generalized Fourier series, including the Fourier–Legendre series and the Fourier–Bessel series. This generalization is as follows.

Let y₀, y₁, y₂, … be orthogonal with respect to a weight function r(x) on an interval a x b, and let f(x) be a function that can be represented by a convergent series

This is called an orthogonal series, orthogonal expansion, or generalized Fourier series. If the y_m are the eigenfunctions of a Sturm–Liouville problem, we call (1) an eigenfunction expansion. In (1) we use again m for summation since n will be used as a fixed order of Bessel functions.

Given f(x), we have to determine the coefficients in (1), called the Fourier constants of f(x) with respect to y₀, y₁…. Because of the orthogonality, this is simple. Similarly to Sec. 11.1, we multiply both sides of (1) by r(x)y_n(x) (n fixed) and then integrate on

both sides from a to b. We assume that term-by-term integration is permissible. (This is justified, for instance, in the case of “uniform convergence,” as is shown in Sec. 15.5.) Then we obtain

Because of the orthogonality all the integrals on the right are zero, except when m = n. Hence the whole infinite series reduces to the single term

Assuming that all the functions y_n have nonzero norm, we can divide by ||y_n||²; writing again m for n, to be in agreement with (1), we get the desired formula for the Fourier constants

This formula generalizes the Euler formulas (6) in Sec. 11.1 as well as the principle of their derivation, namely, by orthogonality.

Hence we have obtained infinitely many orthogonal sets of Bessel functions, one for each of J₀, J₁, J₂, …. Each set is orthogonal on an interval 0 x R with a fixed positive R of our choice and with respect to the weight x. The orthogonal set for J_n is J_n(k_n,1x), J_n(k_n,2x), J_n(k_n,3x), …, where n is fixed and k_n,m is given by (7).

Step 3. Fourier–Bessel series. The Fourier–Bessel series corresponding to J_n (n fixed) is

The coefficients are (with α_n,m = k_n,mR)

because the square of the norm is

as we state without proof (which is tricky; see the discussion beginning on p. 576 of [A13]).

Mean Square Convergence. Completeness

Ideas on approximation in the last section generalize from Fourier series to orthogonal series (1) that are made up of an orthonormal set that is “complete,” that is, consists of “sufficiently many” functions so that (1) can represent large classes of other functions (definition below).

In this connection, convergence is convergence in the norm, also called mean-square convergence; that is, a sequence of functions f_k is called convergent with the limit f if

written out by (5) in Sec. 11.5 (where we can drop the square root, as this does not affect the limit)

Accordingly, the series (1) converges and represents f if

where s_k is the kth partial sum of (1).

Note that the integral in (13) generalizes (3) in Sec. 11.4.

We now define completeness. An orthonormal set y₀, y₁, … on an interval a x b is complete in a set of functions S defined on a x b if we can approximate every f belonging to S arbitrarily closely in the norm by a linear combination a₀y₀ + a₁y₁ + … + a_ky_k, that is, technically, if for every ε > 0 we can find constants a₀, …,a_k (with k large enough) such that

Ref. [GenRef7] in App. 1 uses the more modern term total for complete.

We can now extend the ideas in Sec. 11.4 that guided us from (3) in Sec. 11.4 to Bessel's and Parseval's formulas (7) and (8) in that section. Performing the square in (13) and using (14), we first have (analog of (4) in Sec. 11.4)

The first integral on the right equals because ∫ry_my_l dx = 0 for m ≠ 1, and . In the second sum on the right, the integral equals a_m, by (2) with ||y_m||² = 1. Hence the first term on the right cancels half of the second term, so that the right side reduces to (analog of (6) in Sec. 11.4)

This is nonnegative because in the previous formula the integrand on the left is nonnegative (recall that the weight r(x) is positive!) and so is the integral on the left. This proves the important Bessel's inequality (analog of (7) in Sec. 11.4)

Here we can let k → ∞, because the left sides form a monotone increasing sequence that is bounded by the right side, so that we have convergence by the familiar Theorem 1 in App. A.3.3 Hence

Furthermore, if y₀, y₁, … is complete in a set of functions S, then (13) holds for every f belonging to S. By (13) this implies equality in (16) with k → ∞. Hence in the case of completeness every f in S saisfies the so-called Parseval equality (analog of (8) in Sec. 11.4)

As a consequence of (18) we prove that in the case of completeness there is no function orthogonal to every function of the orthonormal set, with the trivial exception of a function of zero norm:

11.7 Fourier Integral

Fourier series are powerful tools for problems involving functions that are periodic or are of interest on a finite interval only. Sections 11.2 and 11.3 first illustrated this, and various further applications follow in Chap. 12. Since, of course, many problems involve functions that are nonperiodic and are of interest on the whole x-axis, we ask what can be done to extend the method of Fourier series to such functions. This idea will lead to “Fourier integrals.”

In Example 1 we start from a special function f_L of period 2L and see what happens to its Fourier series if we let L → ∞. Then we do the same for an arbitrary function f_L of period 2L. This will motivate and suggest the main result of this section, which is an integral representation given in Theorem 1 below.

Fig. 280. Waveforms and amplitude spectra in Example 1

From Fourier Series to Fourier Integral

We now consider any periodic function f_L(x) of period 2L that can be represented by a Fourier series

and find out what happens if we let L → ∞. Together with Example 1 the present calculation will suggest that we should expect an integral (instead of a series) involving cos wx and sin wx with w no longer restricted to integer multiples w = w_n = nπ/L of π/L but taking all values. We shall also see what form such an integral might have.

If we insert a_n and b_n from the Euler formulas (6), Sec. 11.2, and denote the variable of integration by υ, the Fourier series of f_L(x) becomes

We now set

Then 1/L = Δw/π, and we may write the Fourier series in the form

This representation is valid for any fixed L, arbitrarily large, but finite.

We now let L → ∞ and assume that the resulting nonperiodic function

is absolutely integrable on the x-axis; that is, the following (finite!) limits exist:

Then 1/L→0, and the value of the first term on the right side of (1) approaches zero. Also Δw = π/L → 0 and it seems plausible that the infinite series in (1) becomes an integral from 0 to ∞, which represents f(x), namely,

If we introduce the notations

we can write this in the form

This is called a representation of f(x) by a Fourier integral.

It is clear that our naive approach merely suggests the representation (5), but by no means establishes it; in fact, the limit of the series in (1) as Δw approaches zero is not the definition of the integral (3). Sufficient conditions for the validity of (5) are as follows.

Applications of Fourier Integrals

The main application of Fourier integrals is in solving ODEs and PDEs, as we shall see for PDEs in Sec. 12.6. However, we can also use Fourier integrals in integration and in discussing functions defined by integrals, as the next example.

Fourier Cosine Integral and Fourier Sine Integral

Just as Fourier series simplify if a function is even or odd (see Sec. 11.2), so do Fourier integrals, and you can save work. Indeed, if f has a Fourier integral representation and is even, then B(w) = 0 in (4). This holds because the integrand of B(w) is odd. Then (5) reduces to a Fourier cosine integral

Note the change in A(w): for even f the integrand is even, hence the integral from −∞ to equals ∞ twice the integral from 0 to ∞, just as in (7a) of Sec. 11.2.

Similarly, if f has a Fourier integral representation and is odd, then A(w) = 0 in (4). This is true because the integrand of A(w) is odd. Then (5) becomes a Fourier sine integral

Note the change of B(w) to an integral from 0 to ∞ because B(w) is even (odd times odd is even).

Earlier in this section we pointed out that the main application of the Fourier integral representation is in differential equations. However, these representations also help in evaluating integrals, as the following example shows for integrals from 0 to ∞.

11.8 Fourier Cosine and Sine Transforms

An integral transform is a transformation in the form of an integral that produces from given functions new functions depending on a different variable. One is mainly interested in these transforms because they can be used as tools in solving ODEs, PDEs, and integral equations and can often be of help in handling and applying special functions. The Laplace transform of Chap. 6 serves as an example and is by far the most important integral transform in engineering.

Next in order of importance are Fourier transforms. They can be obtained from the Fourier integral in Sec. 11.7 in a straightforward way. In this section we derive two such transforms that are real, and in Sec. 11.9 a complex one.

Fourier Cosine Transform

The Fourier cosine transform concerns even functions f(x). We obtain it from the Fourier cosine integral [(10) in Sec. 10.7]

Namely, we set , where c suggests “cosine.” Then, writing υ = x in the formula for A(w), we have

and

Formula (1a) gives from f(x) a new function , called the Fourier cosine transform of f (x). Formula (1b) gives us back f(x) from , and we therefore call f(x) the inverse Fourier cosine transform of .

The process of obtaining the transform from a given f is also called the Fourier cosine transform or the Fourier cosine transform method.

Fourier Sine Transform

Similarly, in (11), Sec. 11.7, we set , where s suggests “sine.” Then, writing υ = x we have from (11), Sec. 11.7, the Fourier sine transform, of f(x) given by

and the inverse Fourier sine transform of , given by

The process of obtaining f_s(w) from f(x) is also called the Fourier sine transform or the Fourier sine transform method.

Other notations are

and and for the inverses of and , respectively.

What did we do to introduce the two integral transforms under consideration? Actually not much: We changed the notations A and B to get a “symmetric” distribution of the constant 2/π in the original formulas (1) and (2). This redistribution is a standard convenience, but it is not essential. One could do without it.

What have we gained? We show next that these transforms have operational properties that permit them to convert differentiations into algebraic operations (just as the Laplace transform does). This is the key to their application in solving differential equations.

Linearity, Transforms of Derivatives

If f(x) is absolutely integrable (see Sec. 11.7) on the positive x-axis and piecewise continuous (see Sec. 6.1) on every finite interval, then the Fourier cosine and sine transforms of f exist.

Furthermore, if f and g have Fourier cosine and sine transforms, so does af + bg for any constants a and b, and by (1a)

The right side is . Similarly for , by (2). This shows that the Fourier cosine and sine transforms are linear operations,

Formula (4a) with f′ instead of f gives (when f′, f″ satisfy the respective assumptions for f, f′ in Theorem 1)

hence by (4b)

Similarly,

A basic application of (5) to PDEs will be given in Sec. 12.7. For the time being we show how (5) can be used for deriving transforms.

Tables of Fourier cosine and sine transforms are included in Sec. 11.10.

11.9 Fourier Transform. Discrete and Fast Fourier Transforms

In Sec. 11.8 we derived two real transforms. Now we want to derive a complex transform that is called the Fourier transform. It will be obtained from the complex Fourier integral, which will be discussed next.

Complex Form of the Fourier Integral

The (real) Fourier integral is [see (4), (5), Sec. 11.7]

where

Substituting A and B into the integral for f, we have

By the addition formula for the cosine [(6) in App. A3.1] the expression in the brackets […] equals cos (wx − wυ) or, since the cosine is even, cos (wx − wυ). We thus obtain

The integral in brackets is an even function of w, call it F(w), bacause cos (wx − wυ) is an even function of w the function f does not depend on w, and we integrate with respect to υ (not w). Hence the integral of F(w) from w = 0 to ∞ is times the integral of F(w) from −∞ to ∞. Thus (note the change of the integration limit!)

We claim that the integral of the form (1) with sin instead of cos is zero:

This is true since sin (wx − wυ) is an odd function of w, which makes the integral in brackets an odd function of w, call it G(w). Hence the integral of G(w) from −∞ to ∞ is zero, as claimed.

We now take the integrand of (1) plus times the integrand of (2) and use the Euler formula [(11) in Sec. 2.2]

Taking wx − wυ instead of x in (3) and multiplying by f(υ) gives

Hence the result of adding (1) plus i times (2), called the complex Fourier integral, is

To obtain the desired Fourier transform will take only a very short step from here.

Fourier Transform and Its Inverse

Writing the exponential function in (4) as a product of exponential functions, we have

The expression in brackets is a function of w, is denoted by , and is called the Fourier transform of f; writing υ = x, we have

With this, (5) becomes

and is called the inverse Fourier transform of .

Another notation for the Fourier transform is

so that

The process of obtaining the Fourier transform from a given f is also called the Fourier transform or the Fourier transform method.

Using concepts defined in Secs. 6.1 and 11.7 we now state (without proof) conditions that are sufficient for the existence of the Fourier transform.

Physical Interpretation: Spectrum

The nature of the representation (7) of f(x) becomes clear if we think of it as a superposition of sinusoidal oscillations of all possible frequencies, called a spectral representation. This name is suggested by optics, where light is such a superposition of colors (frequencies). In (7), the “spectral density” measures the intensity of f(x) in the frequency interval between w and w + Δw (Δw small, fixed). We claim that, in connection with vibrations, the integral

can be interpreted as the total energy of the physical system. Hence an integral of from a to b gives the contribution of the frequencies w between a and b to the total energy.

To make this plausible, we begin with a mechanical system giving a single frequency, namely, the harmonic oscillator (mass on a spring, Sec. 2.4)

Here we denote time t by x. Multiplication by y′ gives my′ y″ + ky′y = 0. By integration,

where υ = y′ is the velocity. The first term is the kinetic energy, the second the potential energy, and the total energy of the system. Now a general solution is (use (3) in Sec. 11.4 with t = x)

where . We write simply A = c₁e^iw₀x, B = c₋₁e^−iw₀x. Then y = A + B. By differentiation, υ = y′ = A′ + B′ = iw₀(A − B). Substitution of υ and y on the left side of the equation for E₀ gives

Here , as just stated; hence . Also i² = −1, so that

Hence the energy is proportional to the square of the amplitude |c₁|.

As the next step, if a more complicated system leads to a periodic solution y = f(x) that can be represented by a Fourier series, then instead of the single energy term |c₁|² we get a series of squares |c_n|² of Fourier coefficients c_n given by (6), Sec. 11.4. In this case we have a “discrete spectrum” (or “point spectrum”) consisting of countably many isolated frequencies (infinitely many, in general), the corresponding |c_n|² being the contributions to the total energy.

Finally, a system whose solution can be represented by an integral (7) leads to the above integral for the energy, as is plausible from the cases just discussed.

Linearity. Fourier Transform of Derivatives

New transforms can be obtained from given ones by using

In applying the Fourier transform to differential equations, the key property is that differentiation of functions corresponds to multiplication of transforms by iw:

Two successive applications of (9) give

Since (iw)² = −w² we have for the transform of the second derivative of f

Similarly for higher derivatives.

An application of (10) to differential equations will be given in Sec. 12.6. For the time being we show how (9) can be used to derive transforms.

Convolution

The convolution f * g of functions f and g is defined by

The purpose is the same as in the case of Laplace transforms (Sec. 6.5): taking the convolution of two functions and then taking the transform of the convolution is the same as multiplying the transforms of these functions (and multiplying them by ):

By taking the inverse Fourier transform on both sides of (12), writing and as before, and noting that and 1/ in (12) and (7) cancel each other, we obtain

a formula that will help us in solving partial differential equations (Sec. 12.6).

Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT)

In using Fourier series, Fourier transforms, and trigonometric approximations (Sec. 11.6) we have to assume that a function f(x), to be developed or transformed, is given on some interval, over which we integrate in the Euler formulas, etc. Now very often a function f(x) is given only in terms of values at finitely many points, and one is interested in extending Fourier analysis to this case. The main application of such a “discrete Fourier analysis” concerns large amounts of equally spaced data, as they occur in telecommunication, time series analysis, and various simulation problems. In these situations, dealing with sampled values rather than with functions, we can replace the Fourier transform by the so-called discrete Fourier transform (DFT) as follows.

Let f(x) be periodic, for simplicity of period 2π. We assume that N measurements of f(x) are taken over the interval 0 x 2π at regularly spaced points

We also say that f(x) is being sampled at these points. We now want to determine a complex trigonometric polynomial

that interpolates f(x) at the nodes (14), that is, q(x_k) = f(x_k) written out, with f_k denoting f(x_k)

Hence we must determine the coefficients c₀, …,c_N−1 such that (16) holds. We do this by an idea similar to that in Sec. 11.1 for deriving the Fourier coefficients by using the orthogonality of the trigonometric system. Instead of integrals we now take sums. Namely, we multiply (16) by e^−imx_k (note the minus!) and sum over k from 0 to N − 1. Then we interchange the order of the two summations and insert x_k from (14). This gives

Now

We donote […] by r. For n = w we have r = e⁰ = 1. The sum of these terms over k equals N, the number of these terms. For n ≠ m we have r ≠ 1 and by the formula for a geometric sum [(6) in Sec. 15.1 with q = r and n = N − 1]

because r^N = 1; indeed, since k, m, and n are integers,

This shows that the right side of (17) equals c_mN. Writing n for m and dividing by N, we thus obtain the desired coefficient formula

Since computation of the c_n (by the fast Fourier transform, below) involves successive halfing of the problem size N, it is practical to drop the factor 1/N from c_n and define the discrete Fourier transform of the given signal f = [f₀ … f_N−1]^T to be the vector with components

This is the frequency spectrum of the signal.

In vector notation, , where the N × N Fourier matrix F_N = [e_nk] has the entries [given in (18)]

where n, k = 0, …, N − 1.

From the DFT (the frequency spectrum) we can recreate the given signal , as we shall now prove. Here F_N and its complex conjugate satisfy

where I is the N × N unit matrix; hence F_N has the inverse

We have seen that is the frequency spectrum of the signal f(x). Thus the components of give a resolution of the 2π-periodic function f(x) into simple (complex) harmonics. Here one should use only n's that are much smaller than N/2, to avoid aliasing. By this we mean the effect caused by sampling at too few (equally spaced) points, so that, for instance, in a motion picture, rotating wheels appear as rotating too slowly or even in the wrong sense. Hence in applications, N is usually large. But this poses a problem. Eq. (18) requires O(N) operations for any particular n, hence O(N²) operations for, say, all n < N/2. Thus, already for 1000 sample points the straightforward calculation would involve millions of operations. However, this difficulty can be overcome by the so-called fast Fourier transform (FFT), for which codes are readily available (e.g., in Maple). The FFT is a computational method for the DFT that needs only O(N) log₂N operations instead of O(N²). It makes the DFT a practical tool for large N. Here one chooses N = 2^p (p integer) and uses the special form of the Fourier matrix to break down the given problem into smaller problems. For instance, when N = 1000, those operations are reduced by a factor 1000/log₂ 1000 ≈ 100.

The breakdown produces two problems of size M = N/2. This breakdown is possible because for N = 2M we have in (19)

The given vector f = [f₀ … f_{N − 1}]^T is split into two vectors with M components each, namely, f_ev = [f₀ f₂ … f_N−2]^T containing the even components of f, and f_od = [f₁ f₃ … f_N−1]^T containing the odd components of f. For f_ev and f_od we determine the DFTs

and

involving the same M × M matrix F_M. From these vectors we obtain the components of the DFT of the given vector f by the formulas

For N = 2^p this breakdown can be repeated p − 1 times in order to finally arrive at N/2 problems of size 2 each, so that the number of multiplications is reduced as indicated above.

We show the reduction from N = 4 to M = N/2 = 2 and then prove (22).

We prove (22). From (18) and (19) we have for the components of the DFT

Splitting into two sums of M = N/2 terms each gives

We now use and pull out from under the second sum, obtaining

The two sums are f_ev,n and f_od,n, the components of the “half-size” transforms Ff_ev and Ff_od.

Formula (22a) is the same as (23). In (22b) we have n + M instead of n. This causes a sign changes in (23), namely − before the second sum because

This gives the minus in (22b) and completes the proof.

11.10 Tables of Transforms

Table I. Fourier Cosine Transforms

See (2) in Sec. 11.8.

Table II. Fourier Sine Transforms

See (5) in Sec. 11.8.

Table III. Fourier Transforms

See (6) in Sec. 11.9.

CHAPTER 11 REVIEW QUESTIONS AND PROBLEMS

1. What is a Fourier series? A Fourier cosine series? A half-range expansion? Answer from memory.
2. What are the Euler formulas? By what very important idea did we obtain them?
3. How did we proceed from 2π-periodic to general-periodic functions?
4. Can a discontinuous function have a Fourier series? A Taylor series? Why are such functions of interest to the engineer?
5. What do you know about convergence of a Fourier series? About the Gibbs phenomenon?
6. The output of an ODE can oscillate several times as fast as the input. How come?
7. What is approximation by trigonometric polynomials? What is the minimum square error?
8. What is a Fourier integral? A Fourier sine integral? Give simple examples.
9. What is the Fourier transform? The discrete Fourier transform?
10. What are Sturm–Liouville problems? By what idea are they related to Fourier series?

11–20 FOURIER SERIES. In Probs. 11, 13, 16, 20 find the Fourier series of f(x) as given over one period and sketch f(x) and partial sums. In Probs. 12, 14, 15, 17–19 give answers, with reasons. Show your work detail.

• 11.
• 12. Why does the series in Prob. 11 have no cosine terms?
• 13.
• 14. What function does the series of the cosine terms in Prob. 13 represent? The series of the sine terms?
• 15. What function do the series of the cosine terms and the series of the sine terms in the Fourier series of e^x(−5 < x < 5) represent?
• 16. f(x) = |x| (−π < x < π)
• 17. Find a Fourier series from which you can conclude that 1 − 1/3 + 1/5 − 1/7 + −… = π/4.
• 18. What function and series do you obtain in Prob. 16 by (termwise) differentiation?
• 19. Find the half-range expansions of f(x) = x (0 < x < 1).
• 20. f(x) = 3x² (−π < x < π)

21–22 GENERAL SOLUTION

Solve, y″ + ω²y = r(t), where |ω| ≠ 0, 1, 2, …, r(t) is 2π-periodic and

• 21. r(t) = 3t² (−π < t < π)
• 22. r(t) = |t| (−π < t < π)

23–25 MINIMUM SQUARE ERROR

• 23. Compute the minimum square error for f(x) = x/π (−π < x < π) and trigonometric polynomials of degree N = 1, …, 5.
• 24. How does the minimum square error change if you multiply f(x) by a constant k?
• 25. Same task as in Prob. 23, for f(x) = |x|/π (−π < x < π). Why is E* now much smaller (by a factor 100, approximately!)?

26–30 FOURIER INTEGRALS AND TRANSFORMS

Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.

• 26. f(x) = x + 1 if 0 < x < 1 and 0 otherwise; by the Fourier sine transform
• 27. f(x) = x if 0 < x < 1 and 0 otherwise; by the Fourier integral
• 28. f(x) = kx if a < x < b and 0 otherwise; by the Fourier transform
• 29. f(x) = x if 1 < x < a and 0 otherwise; by the Fourier cosine transform
• 30. f(x) = e^−2x if x > 0 and 0 otherwise; by the Fourier transform

SUMMARY OF CHAPTER 11 Fourier Analysis. Partial Differential Equations (PDEs)

Fourier series concern periodic functions f(x) of period p = 2L, that is, by definition f(x + p) = f(x) for all x and some fixed p > 0; thus, f(x + np) = f(x) for any integer n. These series are of the form

with coefficients, called the Fourier coefficients of f(x), given by the Euler formulas (Sec. 11.2)

where n = 1, 2, …. For period 2π we simply have (Sec. 11.1)

with the Fourier coefficients of f(x) (Sec. 11.1)

Fourier series are fundamental in connection with periodic phenomena, particularly in models involving differential equations (Sec. 11.3, Chap, 12). If f(x) is even [f(−x) = f(x)] or odd [f(−x) = −f(x)], they reduce to Fourier cosine or Fourier sine series, respectively (Sec. 11.2). If f(x) is given for 0 x L only, it has two half-range expansions of period 2L, namely, a cosine and a sine series (Sec. 11.2).

The set of cosine and sine functions in (1) is called the trigonometric system. Its most basic property is its orthogonality on an interval of length 2L; that is, for all integers m and n ≠ m we have

and for all integers m and n,

This orthogonality was crucial in deriving the Euler formulas (2).

Partial sums of Fourier series minimize the square error (Sec. 11.4).

Replacing the trigonometric system in (1) by other orthogonal systems first leads to Sturm–Liouville problems (Sec. 11.5), which are boundary value problems for ODEs. These problems are eigenvalue problems and as such involve a parameter λ that is often related to frequencies and energies. The solutions to Sturm–Liouville problems are called eigenfunctions. Similar considerations lead to other orthogonal series such as Fourier–Legendre series and Fourier–Bessel series classified as generalized Fourier series (Sec. 11.6).

Ideas and techniques of Fourier series extend to nonperiodic functions f(x) defined on the entire real line; this leads to the Fourier integral

where

or, in complex form (Sec. 11.9),

where

Formula (6) transforms f(x) into its Fourier transform , and (5) is the inverse transform.

Related to this are the Fourier cosine transform (Sec. 11.8)

and the Fourier sine transform (Sec. 11.8)

The discrete Fourier transform (DFT) and a practical method of computing it, called the fast Fourier transform (FFT), are discussed in Sec. 11.9.