Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks


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1 Chapter 3 Fourier Series 3.1 Some Properties of Functions Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results will be needed for the remaining sections. We also introduce some notation Preliminary Remarks Joseph Fourier ( ) who gave his name to Fourier series, was not the first to use Fourier series neither did he answer all the questions about them. These series had already been studied by Euler, d Alembert, Bernoulli and others before him. Fourier also thought wrongly that any function could be represented by Fourier series. However, these series bear his name because he studied them extensively. The first concise study of these series appeared in Fourier s publications in 187, 1811 and 1822 in his Théorie analytique de la chaleur. He applied the technique of Fourier series to solve the heat equation. He had the insight to see the power of this new method. His work set the path for techniques that continue to be developed even today. Fourier Series, like Taylor series, are special types of expansion of functions. With Taylor series, we are interested in expanding a function in terms of the special set of functions 1, x, x 2, x 3,... or more generally in terms of 1, (x a), (x a) 2, (x a) 3,... You will remember from calculus that if a function f has a power series representation at a then f (x) = n= f (n) (a) n! (x a) n (3.1) Remember from calculus that a series is an infinite sum. We never use the full se 117
2 118 CHAPTER 3. FOURIER SERIES N f (n) (a) ries, we usually truncate it. In other words, if we call S N (x) = (x a) n, n! n= then we approximate f (x) by S n (x). S n (x) is called a partial sum. A reason for using Taylor series is that their partial sums are polynomials and polynomials are the easiest functions to work with. With Fourier series, we are interested in expanding a function f in terms of the special set of functions 1, cos πx 2πx 3πx πx 2πx, cos, cos,..., sin, sin, sin 3πx,... Thus, a Fourier series expansion of a function is an expression of the form f (x) = A + n=1 ( A n cos nπx + B n sin nπx ) for some positive constant. In the previous chapters, we saw this was useful in helping us to solve certain PDEs. Another reason for using Fourier series is if f (x) represents some signal (light, sound) since signals are a combination of periodic functions. So it is natural we might want to write f (x) as a Fourier series. However, there are several questions which arise when trying to achieve this. We list them here and will try to answer most of them in this chapter. It is important for the reader to be aware of these questions. 1. Given a function f (x), how do we know if it has a Fourier series representation? Fourier thought every function did. It turns out it is not quite the case, though many functions do. 2. Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? We have already answered this question in the previous chapters. 3. Infinite series do not always converge. Some converge for some values of x and not for others. Does the Fourier series converge and for which values of x? 4. Even if the Fourier series of a function f converges, does it converge to f (x)? We will see that even if a Fourier series converges, it does not always converge to f (x). 5. Since a Fourier series is an infinite sum, the sum rules we know for derivatives and integrals do not apply here. In other words, how do we differentiate and integrate a Fourier series? Can we differentiate and integrate a Fourier series term by term? We assumed we could in the previous chapter, but we do not know that. Though we will see it is true for most functions, it is not always true. 6. Given an initial boundary value problem (IBVP), is the resulting Fourier series really a solution of the IBVP? Keep in mind that to
3 3.1. SOME PROPERTIES OF FUNCTIONS 119 check this, we will have to differentiate the Fourier series, hence the answer to the previous question is relevant. After reviewing some properties of functions, we will review how to represent a function by its Fourier series. We will then answer the question listed above. We will finish these notes by discussing some applications of Fourier series Even, Odd Functions Definition 16 (Even and Odd) et f be a function defined on an interval I (finite or infinite) centered at x =. 1. f is said to be even if f ( x) = f (x) for every x in I. 2. f is said to be odd if f ( x) = f (x) for every x in I. The graph of an even function is symmetric with respect to the yaxis. The graph of an odd function is symmetric with respect to the origin. Example 17 1, x 2, x n (where n is even), and cos x are all even functions Example 18 x, x 3, x n (where n is odd), and sin x are all odd functions. You will recall from calculus the following important theorem about integrating even and odd functions over an interval of the form [ a, a] where a >. Theorem 19 et f be a function which domain includes [ a, a] where a >. 1. If f is even, then a a f (x) dx = 2 a f (x) dx 2. If f is odd, then a f (x) dx = a There are several useful algebraic properties of even and odd functions as shown in the theorem below. Theorem 11 When adding or multiplying even and odd functions, the following is true: even + even = even odd + odd = odd even even = even odd odd = even even odd = odd
4 12 CHAPTER 3. FOURIER SERIES Figure 3.1: Graph of an Even Function Figure 3.2: Graph of an Odd Function
5 3.1. SOME PROPERTIES OF FUNCTIONS Periodic Functions Definition 111 (Periodic) et T >. 1. A function f is called T periodic or simply periodic if for all x. 2. The number T is called a period of f. f (x + T ) = f (x) (3.2) 3. If f is nonconstant, then the smallest positive number T with the above property is called the fundamental period or simply the period of f. Remark 112 et us first remark that if T is a period for f, then nt is also a period for any integer n >. This is easy to see using equation 3.2 repeatedly: f (x) = f (x + T ) = f ((x + T ) + T ) = f (x + 2T ) = f ((x + 2T ) + T ) = f (x + 3T ). = f ((x + (n 1) T ) + T ) = f (x + nt ) Classical examples of periodic functions are sin x, cos x and other trigonometric functions. sin x and cos x have period 2π. tan x has period π. We will see more examples below. Because the values of a periodic function of period T repeat every T units, it is enough to know such a function on any interval of length T. Its graph is obtained by repeating the portion over any interval of length T. Consequently, to define a T periodic function, it is enough to define it over any interval of length T. Since different intervals may be chosen, the same function may be defined different ways. Example 113 Describe the 2periodic function shown in figure 3.3 in two different ways: 1. By considering its values on the interval x < 2; 2. By considering its values on the interval 1 x < 1. Solution 1. On the interval x < 2, the function is a portion of the line y = x + 1 thus f (x) = x + 1 if x < 2. The relation f (x + 2) = f (x) describes f for all other values of x. 2. On the interval 1 x < 1, the function consists of two lines. So we have { x 1 if 1 x < f (x) = x + 1 if x, 1 The relation f (x + 2) = f (x) describes f for all other values of x.
6 122 CHAPTER 3. FOURIER SERIES Figure 3.3: A Function of Period 2 Although we have different formulas, they describe the same function. Of course, in practice, we use common sense to select the most appropriate formula. Next, we look at an important theorem concerning integration of periodic functions over one period. Theorem 114 (Integration Over One Period) Suppose that f is T periodic. Then for any real number a, we have T f (x) dx = a+t a f (x) dx (3.3) Proof. Define F (a) = a+t f (x) dx. By the fundamental theorem of calculus, a F (a) = f (a + T ) f (a) = since f is T periodic. Hence, F (a) is a constant for all a. In particular, F () = F (a) which implies the theorem. We illustrate this theorem with an example. Example 115 et f be the 2periodic function shown in figure 3.3. Compute the integrals below: [f (x)]2 dx 2. N N [f (x)]2 dx where N is any positive integer. Solution 116 We answer each part separately. 1. We described this function earlier and noticed that its simplest expression was not over the interval [ 1, 1] but over the interval [, 2]. We should
7 3.1. SOME PROPERTIES OF FUNCTIONS 123 also note that if f is 2periodic, so is [f (x)] 2 (why?). Using theorem 114, we have 1 1 [f (x)] 2 dx = = 2 2 = 1 3 = 2 3 [f (x)] 2 dx ( x + 1) 2 dx ( x + 1) We break N N [f (x)]2 dx into the sum of N integrals over intervals of length 2. N N [f (x)] 2 dx = N+2 N [f (x)] 2 dx+ By theorem 114, each integral is 2 3. Thus N N N+4 N+2 [f (x)] 2 dx = 2N 3 N [f (x)] 2 dx+...+ [f (x)] 2 dx N 2 The following result about combining periodic functions is important. Theorem 117 When combining periodic functions, the following is true: 1. If f 1, f 2,..., f n are T periodic, then a 1 f 1 + a 2 f a n f n is also T  periodic. 2. If f and g are two T periodic functions so is f (x) g (x). 3. If f and g are two T periodic functions so is f(x) g(x) where g (x). 4. If f has period T and a > then f ( x a ) has period at and f (ax) has period T a. 5. If f has period T and g is any function (not necessarily periodic) then the composition g f has period T. Proof. See problems. We finish this section by looking at another example of a periodic function, which does not involve trigonometric functions but rather the greatest integer function, also known as the floor function, denoted x. x represents the greatest integer not larger than x. For example, 5.2 = 5, 5 = 5, 5.2 = 6, 5 = 5. Its graph is shown in figure 3.4.
8 124 CHAPTER 3. FOURIER SERIES Figure 3.4: Graph of x Figure 3.5: Graph of x x Example 118 et f (x) = x x. This gives the fractional part of x. For x < 1, x =, so f (x) = x. Also, since x + 1 = 1 + x, we get f (x + 1) = x + 1 x + 1 = x x = x x = f (x) So, f is periodic with period 1. Its graph is obtained by repeating the portion of its graph over the interval x < 1. Its graph is shown in figure The practice problems will explore further properties of periodic functions.
9 3.1. SOME PROPERTIES OF FUNCTIONS Orthogonal Family of Functions The functions in the 2periodic trigonometric system 1, cos πx, cos 2πx, cos 3πx,..., sin πx 2πx 3πx, sin, sin,... are among the most important periodic functions. The reader will verify that they are indeed 2periodic in the homework. They share another important property. Theorem 119 The family of functions {1, cos nπ x, sin nπ } x : n N forms an orthogonal family on the interval [, ] in other words, if m and n are two nonnegative integers, then 1, cos nπ x = for n = 1, 2,... (3.4) 1, sin nπ x = for n = 1, 2,... sin nπ mπ x, cos x = m, n sin nπ mπ x, sin x = if m n cos nπ mπ x, cos x = if m n Proof. The proof has been done in the previous chapter over several section. We remind the reader of the important trigonometric identities which are used in evaluating these integrals. sin α cos β = 1 [sin (α + β) + sin (α β)] 2 cos α sin β = 1 [sin (α + β) sin (α β)] 2 sin α sin β = 1 [cos (α + β) cos (α β)] 2 cos α cos β = 1 [cos (α + β) + cos (α β)] 2 Remark 12 We also have the useful identities cos 2 mπ xdx = Practice Problems 1. Prove theorem Prove theorem 11. sin 2 mπ xdx = for all m (3.5)
10 126 CHAPTER 3. FOURIER SERIES 3. Sums of periodic functions. Show that if f 1, f 2,..., f n are T periodic, then a 1 f 1 + a 2 f a n f n is also T periodic. 4. Sums of periodic functions. et f (x) = cos x + cos πx. (a) Show that the equation f (x) = 2 has a unique solution. (b) Conclude from part a that f is not periodic. Does this contradict the previous problem? 5. Finish proving theorem Operations on periodic functions. (a) Show that if f and g are two T periodic functions so is f (x) g (x). (b) Show that if f and g are two T periodic functions so is f(x) g(x) g (x). where (c) Show that if f has period T and a > then f ( x a ) has period at and f (ax) has period T a. (d) Show that if f has period T and g is any function (not necessarily periodic) then the composition g f has period T. 7. Using the previous problem, find the period of the functions below. (a) sin 2x (b) cos 1 2x + 3 sin 2x (c) 1 2+sin x (d) e cos x 8. Show that the functions 1, cos πx are 2periodic., cos 2πx, cos 3πx,..., sin πx 2πx 3πx, sin, sin, Antiderivative of periodic functions. Suppose that f is 2πperiodic and let a be a fixed real number. Define F (x) = x a f (t) dt for all x Show that F is 2πperiodic if and only if 2π f (t) dt =. (hint: use theorem 114)
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