1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
|
#<b>Why Fourier series possible?</b>
We first begin with a few basic identities on the size of sets. Show that the set of possible functions representing sets is not larger than the set of available functions?
#<b>Why Fourier series is plausible?</b>
To show that Fourier series is plausible, let us consider some arbitrary trignometric functions and see if it is possible to express them as the sum of sines and cosines:
$1.\quad\sin^2(x) = ?$
Based on the double angle formula,
$$\cos(2x) = 1 - 2 \sin^2(x)$$
Rearranging,
$$\sin^2(x) = \frac{1-\cos(2x)}{2}$$
$2.\quad\sin(2x)\cdot\cos(2x) = ?$
Based on the double angle formula,
$$\qquad\sin(2x) = 2\sin(x)\cos(x)$$
Rearranging,
$$\begin{array}{ccl}
\sin(2x)\cdot\cos(x) & = & [2\sin(x)\cos(x)]\cdot\cos(x)\\
& = & 2 \sin(x) [ 1 - \sin^2(x)]\\
& = & 2\sin(x) - 2\sin^3(x)\\
\end{array}$$
Based on the triple angle formula,
$$\sin(3x) = 3\sin(x) - 4\sin^3(x)$$
Rearranging,
$$\sin^3(x) = \frac{3\sin(x)-\sin(3x)}{4}$$
Substituting back in the former equation, we get
$$
\begin{array} {ccl}
\sin(2x) & = & 2\sin(x) - 2 [\frac{3\sin(x)-\sin(3x)}{4}]\\
& = & \frac{1}{2}\sin(x) + \frac{1}{2}\sin(3x)\\
\end{array}
$$
Thus, we see that both these functions could be expressed as sums of sines and cosines. It is possible to show that every product of trignometric functions can be expressed as a sum of sines and cosines:
$$
\begin{array}{ccl}
e^{i\theta} & = & \cos \theta + i \sin \theta\\
e^{-i\theta} & = & \cos \theta - i \sin \theta\\
\end{array}{ccl}
$$
Solving for $\cos \theta$ and $\sin \theta$
$$
\begin{array}{ccl}
\cos \theta & = & \frac{1}{2}e^{i\theta} + \frac{1}{2}e^{-i\theta}\\
\sin \theta & = & \frac{1}{2i}e^{i\theta} - \frac{1}{2i}e^{-i\theta}\\
\end{array}
$$
It is easy to show that any product of cosines and sines can be expressed as the product of exponentials which will reduce to a sum of sines and cosines.
As a final test to see if the Fourier series really could exist for any periodic function, we consider a periodic function with a sharp peak as shown below
<center> </center>
If it is possible to approximate the above function using a sum of sines and cosines, then it can be argued that *any* continuous periodic function can be expressed in a similar way. This is because any function could be expressed as a number of peaks at every position.
It turns out that the above function can be approximated as the difference of two cosines, namely, $\cos^{2n}(x) + cos^{2n+1}(x)$
<center>  </center>
Summing these two functions we get the following:
<center></center>
#<b>What is the Fourier Series actually?</b>
##Initial Hypothesis
Now, to prove that the Fourier series is indeed true, we begin with the following hypothesis:
Let $f : \mathbb I \rightarrow \mathbb C$ be a continuous, periodic function where $I$ is some time interval(period of the function). Then it can be expressed as :
$$
\begin{array}{ccl}
f & = & \Sigma e^{inx}\\
& = & a_0 + \Sigma a_n\cos nx + \Sigma b_n\sin nx\\
\end{array}
$$
##Definition of Inner Product of Functions
We begin proving this hypothesis by considering that any function on the right-hand side of our hypothesis is uniquely defined by the co-efficients of the terms $a_0$ through $a_n$ and $b_1$ through $b_n$. This can be taken to mean that every function is really a vector in a $2n+1$-dimensional 'Hilbert space'.(*perhaps someone can clarify this?*)
We now proceed to define certain operations on these functions in Hilbert space. One operation that will be particularly useful is that of the inner product of two functions:
$$
\begin{array}{ccl}
inner product, (f,g) & = & \int_0^{2\pi} f \, g \,dx\\
\mid f \mid ^2 = (f, f) & = & \int_0^{2\pi} f^2 \,dx\\
\end{array}
$$
This is the inner product of 2 real-number functions. For a function on complex numbers, the above definition must be altered as follows:
$$
\begin{array}{ccl}
inner product, (f,g) & = & \int_0^{2\pi} f \,\bar g \,dx\\
\mid f \mid ^2 = (f, f) & = & \int_0^{2\pi} f^2 \,dx\\
\end{array}
$$
*Note: These are purely definitions, and we are defining the inner product to ensure that the inner product of f and f is a real number.*
##Basis Vectors of the Hilbert Space
The basis vectors of this Hilbert space are taken as follows:
basis vectors, $f_n = \frac{1}{\sqrt{2\pi}}e^{inx}$
Any basis vectors could conceivable have been assumed on the condition that the basis vectors are orthonormal. (*Note: These particular basis vectors are chosen to prove that Fourier series exists*)
In order to prove orthonormality of the basis vectors:
$$
\begin{array}{ccl}
(f_n,f_m) & = & \int_0^{2\pi} \, \frac{1}{\sqrt{2\pi}} \, e^{inx} \, \bar {\frac{1}{\sqrt{2\pi}} \, e^{inx}} \, dx\\
& = & \frac{1}{2\pi} \, \int_0^{2\pi} \, e^{i(n-m)x} \, dx \\
\end{array}
$$
The exponential can be expanded using $e^{ikx} = \cos kx + i \sin kx$. Then, integrating, we get the following:
$$
\begin{array}{ccl}
n = m \Rightarrow (f_n,f_m) & = & 1\\
n \neq m \Rightarrow (f_n,f_m) & = & 0\\
\end{array}
$$
which is the condition of orthonormality (the vectors are perpendicular and each has a length of 1 unit).
##Determining Coefficients of the Basis vectors
In any vector space, the inner product of a vector and its basis vector gives the coefficient. For example, consider a 2-dimensional vector as shown below:
[Graph of a vector](/vector.gif)
The above vector $\vec v$ can be expressed in terms of the basis vectors $\vec e_1$ and $\vec e_2$ as follows:
$$
\begin{array}{ccl}
\vec v & = & a_1 \, \vec e_1 + a_2 \, \vec e_2\\
& = & a_1 \, \left( \begin{array}{c}1\\0\end{array} \right) + a_2 \, \left( \begin{array}{c}0\\1\end{array} \right)\\
& = & \left( \begin{array}{c}a_1\\a_2\end{array} \right) \\
\end{array}
$$
So,
$$
\begin{array}{ccl}
\vec v . \vec e_1 & = & a_1\\
\vec v . \vec e_2 & = & a_2\\
\end{array}
$$
Extending this principle to the case of an n-dimensional vector:
--> compute inner product here and then continue to show what the coefficient formula is
##Proving that this function is does indeed completely represent $f$
--> don't quite remember this part
#<b>Why is Fourier series useful? </b>
Applications will be covered on Monday July 5, 2010. See you all soon!
|
