11.1 ORTHOGONAL FUNCTIONS

398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES

11.1

ORTHOGONAL FUNCTIONS

REVIEW MATERIAL The notions of generalized vectors and vector spaces can be found in any linear algebra text.

INTRODUCTION The concepts of geometric vectors in two and three dimensions, orthogonal or perpendicular vectors, and the inner product of two vectors have been generalized. It is perfectly routine in mathematics to think of a function as a vector. In this section we will examine an inner product that is different from the one you studied in calculus. Using this new inner product, we define orthogonal functions and sets of orthogonal functions. Another topic in a standard calculus course is the expansion of a function f in a power series. In this section we will also see how to expand a suitable function f in terms of an infinite set of orthogonal functions.

INNER PRODUCT Recall that if u and v are two vectors in 3-space, then the inner product (u, v) (in calculus this is written as u v) possesses the following properties:

(i) (u, v) (v, u), (ii) (ku, v) k(u, v), k a scalar, (iii) (u, u) 0 if u 0 and (u, u) 0 if u 0, (iv) (u v, w) (u, w) (v, w).

We expect that any generalization of the inner product concept should have these same properties.

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

DEFINITION 11.1.1 Inner Product of Functions

The inner product of two functions f1 and f2 on an interval [a, b] is the number

b

( f1, f 2) f1(x) f 2(x) dx.

a

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

DEFINITION 11.1.2 Orthogonal Functions

Two functions f1 and f2 are orthogonal on an interval [a, b] if

b

( f1, f 2) f1(x) f 2(x) dx 0.

(1)

a

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

11.1 ORTHOGONAL FUNCTIONS 399

For example, the functions f1(x) x2 and f2(x) x3 are orthogonal on the interval [1, 1], since

( f1, f2)

1 x2 x3 dx 1 x6 1 0.

1

6 1

Unlike in vector analysis, in which 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.

DEFINITION 11.1.3 Orthogonal Set

A set of real-valued functions {f 0(x), f1(x), f2(x), . . . } is said to be orthogonal on an interval [a, b] if

b

(m, n) m(x)n(x) dx 0, m Y n.

(2)

a

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 1(u, u). Similarly, the square norm of a function fn is fn(x)2 (fn , fn), and so the norm, or its generalized length, is fn(x) 1(n, n). In other words, the square norm and norm of a function fn in

an orthogonal set {fn(x)} are, respectively,

b

fn(x)2 n2 (x) dx

a

and

b

fn(x)

B

f2n(x) dx.

a

(3)

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

EXAMPLE 1 Orthogonal Set of Functions

Show that the set {1, cos x, cos 2x, . . .} is orthogonal on the interval [p, p].

SOLUTION If we make the identification f0(x) 1 and fn(x) cos nx, we must then show that 0(x) n(x) dx 0, n 0, and m(x) n(x) dx 0, m n.

We have, in the first case,

(0 , n) 0(x) n(x) dx cos nx dx

1 sin nx 1 [sin n sin(n)] 0,

n

n

n 0,

400 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES

and, in the second,

(m , n) m(x)n(x) dx

cos mx cos nx dx

1

[cos(m n)x cos(m n)x] dx

2

1 2

sin

(m n) mn

x

sin (m n)x mn

0,

; trig identity

m n.

EXAMPLE 2 Norms

Find the norm of each function in the orthogonal set given in Example 1.

SOLUTION

For f0(x) 1 we have, from (3),

f0 (x)2 dx 2,

so f0(x) 12. For fn(x) cos nx, n 0, it follows that

fn (x)2

cos2 nx dx 1

2

[1 cos 2nx] dx .

Thus for n 0, fn(x) 1.

Any orthogonal set of nonzero functions {fn(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 1 and 2 that the set

1 cos x cos 2x

,,

,...

12 1 1

is orthonormal on the interval [p, p]. 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 ci , i 1, 2, 3, are scalars called the components of the vector. Each component ci can be expressed in terms of u and the corresponding vector vi . 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

c1

(u, v1) 'v1'2

.

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

c2

(u, v2) 'v2'2

and

c3

(u, v3) 'v3'2

.

11.1 ORTHOGONAL FUNCTIONS 401

Hence (4) can be expressed as

u

(u, v1) 'v1'2

v1

(u, v2) 'v2'2

v2

(u, v3) 'v3'2

v3

3 n1

(u, vn ) 'vn'2

vn.

(5)

ORTHOGONAL SERIES EXPANSION Suppose {fn(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 cn, n 0, 1, 2, . . . , for which

f (x) c00(x) c11(x) cnn(x) ?

(6)

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

b

b

b

b

f (x)m(x) dx c0 0(x)m(x) dx c1 1(x)m(x) dx cn n(x)m(x) dx

a

a

a

a

c0(0, m) c1(1, m) cn(n, m) .

By orthogonality each term on the right-hand side of the last equation is zero except

when m n. In this case we have

b

b

f (x) n(x) dx cn 2n(x) dx.

a

a

It follows that the required coefficients are

cn

b a

f

(x) n(x)

dx,

b a

2 n

(x)

d

x

n 0, 1, 2, . . . .

In other words,

f (x) cn n(x),

(7)

n0

where

cn

b a

f (x) n(x) 'n (x)'2

d

x

.

(8)

With inner product notation, (7) becomes

f (x)

n0

( f, n) 'n (x)'2

n (x).

(9)

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

DEFINITION 11.1.4 Orthogonal Set/Weight Function

A set of real-valued functions {f0(x), f1(x), f2(x), . . .} is said to be orthogonal with respect to a weight function w(x) on an interval [a, b] if

b w(x) m(x) n(x) dx 0, m n.

a

The usual assumption is that w(x) 0 on the interval of orthogonality [a, b].

The set {1, cos x, cos 2x, . . .} in Example 1 is orthogonal with respect to the weight

function w(x) 1 on the interval [p, p].

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

cn

b a

f (x) w(x) n(x) 'n (x)'2

dx ,

(10)

402 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES

where

b

fn(x)2 w(x) n2(x) dx.

(11)

a

The series (7) with coefficients 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 cn was formal; that is, basic questions about whether or not 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 fn of the orthogonal set {fn(x)}. (If f were orthogonal to every fn, then cn 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 function that is orthogonal to each member of the set is the zero function.

EXERCISES 11.1

Answers to selected odd-numbered problems begin on page ANS-18.

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) x 3, f2(x) x2 1; [1, 1] 3. f1(x) ex, f2(x) xex ex ; [0, 2] 4. f1(x) cos x, f2(x) sin2x ; [0, p] 5. f1(x) x, f2(x) cos 2x ; [p 2, p 2] 6. f1(x) ex, f2(x) sin x; [p 4, 5p 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, p2] 8. {cos x, cos 3x, cos 5x, . . .}; [0, p 2] 9. {sin nx}, n 1, 2, 3, . . . ; [0, p]

10.

n sin x

, n 1, 2, 3, . . . ;

[0, p]

p

11.

1,

cos

n x

,

n

1,

2,

3,

.

.

.

;

[0, p]

p

12.

n m 1, cos x, sin x

,

n 1, 2, 3, . . . ,

p

p

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) ex2, (, )

14. L 0(x) 1,

L 1(x) x 1,

L 2 (x)

1 2

x2

2x

1;

w(x) ex, [0, )

15.

Let {fn(x)} be an orthogonal such that f0(x) 1. Show

set of functions

that ban(x) dx

on

[a, b] 0 for

n 1, 2, . . . .

16. Let {fn(x)} be an orthogonal set of functions on [a, b]

such

b a

(

x

that f 0(x) )n(x) dx

1

and 0 for

f1(x) x. Show n 2, 3, . . . and

that any

constants a and b.

17. Let {fn(x)} be an orthogonal set of functions on [a, b]. Show that fm(x) fn(x)2 fm(x)2 fn(x)2,

m n.

18. From Problem 1 we know that f1(x) x and f2(x) x 2 are orthogonal on the interval [2, 2]. Find constants c1 and c2 such that f3(x) x c1x2 c2 x3 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 [p, p]. 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).

Discussion Problems

21. A real-valued function f is said to be periodic with period

T if f (x T ) f (x). For example, 4p is a period of sin x,

since sin(x 4p) sin x. The smallest value of T for

which f (x T ) f (x) holds is called the fundamental

period of f. For example, the fundamental period of

f (x) sin x is T 2p. What is the fundamental period

of each of the following functions?

(a) f (x) cos 2px

(b) f (x) sin 4 x L

(c) f (x) sin x sin 2x (d) f (x) sin 2x cos 4x

(e) f (x) sin 3x cos 2x

n

n

(f) f (x) A0

n1

An cos

p

x Bn sin

p

x,

An and Bn depend only on n

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