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
2 3 For example, the functions f1(x) x and f2(x) x are orthogonal on the interval [ 1, 1], since
1 1 1 2 3 6 ( f1, f2) x x dx x 0. 1 6 1
Unlike in vector analysis, in which the word orthogonal is a synonym for perpendic- ular, 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 Y ( m, n) m(x) n(x) dx 0, m 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) u 2 is called the square norm, and so the norm is u 1(u, u). Similarly, the square norm of a function fn 2 is fn(x) (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 b 2 2 2 fn(x) n (x) dx and fn(x) fn(x) dx. (3) a B a
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,