INTRODUCTION

In certain areas of advanced mathematics, a function is considered to be a generalization of a vector. In this section we shall see how the two vector concepts of inner, or dot, product and orthogonality of vectors can be extended to functions. The remainder of the chapter is a practical application of this discussion.

Inner Product

Recall, if u = u₁i + u₂ j + u₃k and v = v₁i + v₂ j + v₃k are two vectors in R³ or 3-space, then the inner product or dot product of u and v is a real number, called a scalar, defined as the sum of the products of their corresponding components:

The inner product (u, v) possesses the following properties:

1. (u, v) = (v, u)
2. (ku, v) = k(u, v), k a scalar
3. (u, u) = 0 if u = 0 and (u, u) > 0 if u ≠ 0
4. (u + v, w) = (u, w) + (v, w).

We expect any generalization of the inner product to possess these same properties.

Suppose that f₁ and f₂ are piecewise-continuous functions defined on an interval [a, b].* Since a definite integral on the interval of the product f₁(x) f₂(x) possesses properties (i)–(iv) of the inner product of vectors, whenever the integral exists we are prompted to make the following definition.

Orthogonal Functions

Motivated by the fact that two vectors u and v are orthogonal whenever their inner product is zero, we define orthogonal functions in a similar manner.

EXAMPLE 1 Orthogonal Functions

Unlike vector analysis, where the word orthogonal is a synonym for perpendicular, in this present context the term orthogonal and condition (1) have no geometric significance.

Orthogonal Sets

We are primarily interested in infinite sets of orthogonal functions.

Orthonormal Sets

The norm, or length u, of a vector u can be expressed in terms of the inner product. The expression (u, u) = u² is called the square norm, and so the norm is u = . Similarly, the square norm of a function ø_n is ø_n(x)² = (ø_n, ø_n), and so the norm, or its generalized length, is ø_n(x) = . In other words, the square norm and norm of a function ø_n in an orthogonal set {ø_n(x)} are, respectively,

(3)

If {ø_n(x)} is an orthogonal set of functions on the interval [a, b] with the property that ø_n(x) = 1 for n = 0, 1, 2, …, then {ø_n(x)} is said to be an orthonormal set on the interval.

EXAMPLE 2 Orthogonal Set of Functions

EXAMPLE 3 Norms

Find the norms of each function in the orthogonal set given in Example 2.

SOLUTION

For ø₀(x) = 1 we have from (3)

ø₀(x)² = dx = 2π

so that ø₀(x) = . For ø_n(x) = cos nx, n > 0, it follows that

Thus for n > 0, ø_n(x) = .≡

Any orthogonal set of nonzero functions {ø_n(x)}, n = 0, 1, 2, …, can be normalized—that is, made into an orthonormal set—by dividing each function by its norm. It follows from Examples 2 and 3 that the set

is orthonormal on the interval [–π, π].

Vector Analogy

We shall make one more analogy between vectors and functions. Suppose v₁, v₂, and v₃ are three mutually orthogonal nonzero vectors in 3-space. Such an orthogonal set can be used as a basis for 3-space; that is, any three-dimensional vector can be written as a linear combination

u = c₁v₁ + c₂v₂ + c₃v₃, (4)

where the c_i , i = 1, 2, 3, are scalars called the components of the vector. Each component c_i can be expressed in terms of u and the corresponding vector v_i . To see this we take the inner product of (4) with v₁:

(u, v₁) = c₁(v₁, v₁) + c₂(v₂, v₁) + c₃(v₃, v₁) = c₁v₁² + c₂ ⋅ 0 + c₃ ⋅ 0.

Hence

In like manner we find that the components c₂ and c₃ are given by

Hence (4) can be expressed as

(5)

Orthogonal Series Expansion

Suppose {ø_n(x)} is an infinite orthogonal set of functions on an interval [a, b]. We ask: If y = f(x) is a function defined on the interval [a, b], is it possible to determine a set of coefficients c_n , n = 0, 1, 2, …, for which

f(x) = c₀ø₀(x) + c₁ø₁(x) + + c_nø_n(x) + ?(6)

As in the foregoing discussion on finding components of a vector, we can find the coefficients c_n by utilizing the inner product. Multiplying (6) by ø_m(x) and integrating over the interval [a, b] gives

By orthogonality, each term on the right-hand side of the last equation is zero except when m = n. In this case we have

f(x)ø_n(x) dx = c_n ø_n² (x) dx.

It follows that the required coefficients c_n are given by

.

In other words, (7)

where (8)

With inner product notation, (7) becomes

(9)

Thus (9) is seen to be the function analogue of the vector result given in (5).

The usual assumption is that w(x) > 0 on the interval of orthogonality [a, b]. The set {1, cos x, cos 2x, …} in Example 2 is orthogonal with respect to the weight function w(x) = 1 on the interval [–π, π].

If {ø_n(x)} is orthogonal with respect to a weight function w(x) on the interval [a, b], then multiplying (6) by w(x)ø_n(x) and integrating yields

(10)

where (11)

The series (7) with coefficients c_n given by either (8) or (10) is said to be an orthogonal series expansion of f or a generalized Fourier series.

Complete Sets

The procedure outlined for determining the coefficients c_n was formal; that is, basic questions on whether an orthogonal series expansion such as (7) is actually possible were ignored. Also, to expand f in a series of orthogonal functions, it is certainly necessary that f not be orthogonal to each ø_n of the orthogonal set {ø_n(x)}. (If f were orthogonal to every ø_n, then c_n = 0, n = 0, 1, 2, ….) To avoid the latter problem we shall assume, for the remainder of the discussion, that an orthogonal set is complete. This means that the only continuous function orthogonal to each member of the set is the zero function.

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

In Problems 1–6, show that the given functions are orthogonal on the indicated interval.

1. f₁(x) = x, f₂(x) = x²; [–2, 2]
2. f₁(x) = x³, f₂(x) = x² + 1; [–1, 1]
3. f₁(x) = e^x, f₂(x) = xe^–x – e^–x; [0, 2]
4. f₁(x) = cos x, f₂(x) = sin² x; [0, π]
5. f₁(x) = x, f₂(x) = cos 2x; [–π/2, π/2]
6. f₁(x) = e^x, f₂(x) = sin x; [π/4, 5π/4]

In Problems 7–12, show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set.

7. {sin x, sin 3x, sin 5x, …}; [0, π/2]
8. {cos x, cos 3x, cos 5x, …}; [0, π/2]
9. {sin nx}, n = 1, 2, 3, …; [0, π]
10. , n = 1, 2, 3, …; [0, p]
11. , n = 1, 2, 3, …; [0, p]
12. , n = 1, 2, 3, …,
    m = 1, 2, 3, …; [–p, p]

In Problems 13 and 14, verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval.

13. H₀(x) = 1, H₁(x) = 2x, H₂(x) = 4x² – 2; w(x) = , (–∞, ∞)
14. L₀(x) = 1, L₁(x) = –x + 1, L₂(x) = x² – 2x + 1; w(x) = e^–x, [0, ∞)
15. Let {ø_n(x)} be an orthogonal set of functions on [a, b] such that ø₀(x) = 1. Show that ø_n(x) dx = 0 for n = 1, 2, ….
16. Let {ø_n(x)} be an orthogonal set of functions on [a, b] such that ø₀(x) = 1 and ø₁(x) = x. Show that (αx + β)ø_n(x) dx = 0 for n = 2, 3, … and any constants α and β.
17. Let {ø_n(x)} be an orthogonal set of functions on [a, b]. Show that ø_m(x) + ø_n(x)² = ø_m(x)² + ø_n(x)², m ≠ n.
18. From Problem 1 we know that f₁(x) = x and f₂(x) = x² are orthogonal on [–2, 2]. Find constants c₁ and c₂ such that f₃(x) = x + c₁x² + c₂x³ is orthogonal to both f₁ and f₂ on the same interval.
19. The set of functions {sin nx}, n = 1, 2, 3, …, is orthogonal on the interval [–π, π]. Show that the set is not complete.
20. Suppose f₁, f₂, and f₃ are functions continuous on the interval [a, b]. Show that (f₁ + f₂, f₃) = (f₁, f₃) + (f₂, f₃).

A real-valued function is said to be periodic with period if for all x in the domain of f. If T is the smallest positive value for which holds, then T is called the fundamental period of f. In Problems 21–26, determine the fundamental period T of the given function.

Discussion Problems

27. The Gram–Schmidt process for constructing an orthogonal set that was discussed in Section 7.7 carries over to a linearly independent set { f₀(x), f₁(x), f₂(x), …} of real-valued functions continuous on an interval [a, b]. With the inner product , define the functions in the set to be

    

    and so on.

    1. Write out ø₃(x) in the set.
    2. By construction, the set is orthogonal on [a, b]. Demonstrate that ø₀(x), ø₁(x), and ø₂(x) are mutually orthogonal.
28. 1. Consider the set of functions {1, x, x², x³, …} defined on the interval [–1, 1]. Apply the Gram–Schmidt process given in Problem 27 to this set and find ø₀(x), ø₁(x), ø₂(x), and ø₃(x) of the orthogonal set B′.
    2. Discuss: Do you recognize the orthogonal set?
29. Verify that the inner product (f₁, f₂) in Definition 12.1.1 satisfies properties (i)–(iv) given on page 681.
30. In R³, give an example of a set of orthogonal vectors that is not complete. Give a set of orthogonal vectors that is complete.
31. The function

    

    where the coefficients A_n and B_n depend only on n, is periodic. Find the period T of f.

*The interval could also be (–∞, ∞), [0, ∞), and so on.