Fourier Series. Some Properties of Functions. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Fourier Series Today 1 / 19

Transcription

1 Fourier Series Some Properties of Functions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 19

2 Introduction 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. Fourier series are named after Joseph Fourier ( ), though he was not the first to use Fourier series neither did he answer all the questions about them. Fourier Series, like Taylor series, are special types of expansion of functions. Philippe B. Laval (KSU) Fourier Series Today 2 / 19

3 Introduction Recall, the Taylor series of a function f is f (n) (a) f (x) = (x a) n, it is valid for x in (a R, a + R) where n! n=0 R is called the radius of convergence. The Fourier series of a function f is ( f (x) = A 0 + A n cos nπx L + B n sin nπx ), it is valid in ( L, L). L n=1 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. Philippe B. Laval (KSU) Fourier Series Today 3 / 19

4 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

5 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? 2 Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

6 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? 2 Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? 3 Does the Fourier series converge and for which values of x? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

7 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? 2 Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? 3 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)? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

8 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? 2 Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? 3 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)? 5 How do we differentiate and integrate a Fourier series? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

9 Some Questions 1 Given a function f (x), how do we know if it has a Fourier series representation? 2 Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? 3 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)? 5 How do we differentiate and integrate a Fourier series? 6 Given an initial boundary value problem (IBVP), is the resulting Fourier series really a solution of the IBVP? Philippe B. Laval (KSU) Fourier Series Today 4 / 19

10 Even and Odd Functions Definition (Even and Odd) Let f be a function defined on an interval I (finite or infinite) centered at x = 0. 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 y-axis. The graph of an odd function is symmetric with respect to the origin. Example 1, x 2, x n (where n is even), and cos x are all even functions Example x, x 3, x n (where n is odd), and sin x are all odd functions. Philippe B. Laval (KSU) Fourier Series Today 5 / 19

11 Even and Odd Functions Figure: Graph of an Even Function Philippe B. Laval (KSU) Fourier Series Today 6 / 19

12 Even and Odd Functions Figure: Graph of an Odd Function Philippe B. Laval (KSU) Fourier Series Today 7 / 19

13 Even and 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 > 0. Theorem Let f be a function which domain includes [ a, a] where a > 0. 1 If f is even, then a a f (x) dx = 2 a 0 f (x) dx 2 If f is odd, then a a f (x) dx = 0 Philippe B. Laval (KSU) Fourier Series Today 8 / 19

14 Even and Odd Functions There are several useful algebraic properties of even and odd functions as shown in the theorem below. Theorem 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 Philippe B. Laval (KSU) Fourier Series Today 9 / 19

15 Periodic Functions Definition (Periodic) Let T > 0. 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) (1) 3 If f is non-constant, then the smallest positive number T with the above property is called the fundamental period or simply the period of f. Remark Let us first remark that if T is a period for f, then nt is also a period for any integer n > 0. This is easy to see using equation 1 repeatedly Philippe B. Laval (KSU) Fourier Series Today 10 / 19

16 Periodic Functions 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. Example Describe the 2-periodic function shown below in two different ways: 1 By considering its values on the interval 0 x < 2; 2 By considering its values on the interval 1 x < 1. Figure: A Function of Period 2 Philippe B. Laval (KSU) Fourier Series Today 11 / 19

17 Periodic Functions Theorem (Integration Over One Period) Suppose that f is T -periodic. Then for any real number a, we have T 0 f (x) dx = a+t a We illustrate this theorem with an example. Example f (x) dx (2) Let f be the 2-periodic function shown in the figure above. Compute the integrals below: [f (x)]2 dx 2 N N [f (x)]2 dx where N is any positive integer. Philippe B. Laval (KSU) Fourier Series Today 12 / 19

18 Periodic Functions The following result about combining periodic functions is important. Theorem 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) 0. 4 If f has period T and a > 0 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. Philippe B. Laval (KSU) Fourier Series Today 13 / 19

19 Periodic Functions 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 below. Figure: Graph of x Philippe B. Laval (KSU) Fourier Series Today 14 / 19

20 Periodic Functions Example Let f (x) = x x. Show f is periodic with period 1. Its graph is shown below. Figure: Graph of x x Philippe B. Laval (KSU) Fourier Series Today 15 / 19

21 Orthogonal Family of Functions The functions in the 2L-periodic trigonometric system 1, cos πx L, cos 2πx L, cos 3πx L,..., sin πx L 2πx 3πx, sin, sin L L,... are among the most important periodic functions. The reader will verify that they are indeed 2L-periodic in the homework. They share another important property. Philippe B. Laval (KSU) Fourier Series Today 16 / 19

22 Orthogonal Family of Functions Theorem The family of functions {1, cos nπ L x, sin nπ } L x : n N forms an orthogonal family on the interval [ L, L] in other words, if m and n are two nonnegative integers, then 1, cos nπ L x = 0 for n = 1, 2,... (3) 1, sin nπ L x = 0 for n = 1, 2,... sin nπ mπ x, cos L L x = 0 m, n sin nπ mπ x, sin L L x = 0 if m n cos nπ mπ x, cos L L x = 0 if m n Philippe B. Laval (KSU) Fourier Series Today 17 / 19

23 Orthogonal Family of Functions 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 We also have the useful identities L L cos 2 mπ L xdx = L L sin 2 mπ xdx = L for all m 0 (4) L Philippe B. Laval (KSU) Fourier Series Today 18 / 19

24 Exercises See the problems at the end of my notes on Fourier series: some properties of functions. Philippe B. Laval (KSU) Fourier Series Today 19 / 19