APPLIED MATHEMTATICS 364
TERM 3
,FOUNDATIONS OF FOURIER ANALYSIS
The main idea we are building towards is:
A complicated function can be represented using simpler basis functions.
Eventually, those basis functions will be sines and cosines, giving us a Fourier series.
We already know how to break a vector into components along different directions.
Fourier analysis essentially does the same thing with functions.
The Vector–Function Analogy
Consider a vector
𝑎1
𝑎
𝑎 = [ 2 ].
𝑎3
A vector is essentially a collection of components.
Now imagine sampling a function 𝑓(𝑥) at discrete points:
𝑥 = 0, ℎ, 2ℎ, 3ℎ, …
We could represent the function values as
𝑓(0)
𝑓(ℎ)
𝑓 = 𝑓(2ℎ) .
𝑓(3ℎ)
[ ⋮ ]
As ℎ → 0, we sample the function at more and more points. In the limit, the function can
be thought of as an infinite-dimensional vector.
,This is important because if functions behave like vectors, we can borrow vector
concepts such as:
• dot products;
• length/norm;
• orthogonality;
• projections;
• basis vectors.
These become:
• inner products;
• function norms;
• orthogonal functions;
• projection onto basis functions;
• basis functions.
And projection onto basis functions is exactly what eventually gives us Fourier
coefficients.
Dot Product → Inner Product
For vectors,
𝑛
𝑎 ⋅ 𝑏 = ∑ 𝑎𝑗 𝑏𝑗 .
𝑗=1
The dot product tells us how much two vectors point in the same direction.
The function equivalent is the inner product.
For two real functions 𝑓(𝑥)and 𝑔(𝑥)on [𝑎, 𝑏],
𝑏
⟨𝑓, 𝑔⟩ = ∫ 𝑓 (𝑥)𝑔(𝑥) 𝑑𝑥
𝑎
Vectors Functions
𝒂⋅𝒃 ⟨𝑓, 𝑔⟩
sum components integrate functions
measures overlap/alignment measures overlap between functions
Function Norm
For a vector, its length is
, ∥ 𝑎 ∥= √𝑎 ⋅ 𝑎.
We define the length or norm of a function in exactly the same way:
∥ 𝑓 ∥= √⟨𝑓, 𝑓⟩
Therefore,
𝑏
∥ 𝑓 ∥= √∫ [ 𝑓(𝑥)]2 𝑑𝑥
𝑎
for real functions.
Normalised Functions
For a vector,
𝑎
𝑢𝑎 =
∥𝑎∥
creates a unit vector (its length is now one)
Similarly, a function can be normalised:
𝜙(𝑥)
𝜙norm (𝑥) =
∥𝜙∥
so that
∥ 𝜙norm ∥= 1.
This distinction becomes important when we talk about orthogonal vs orthonormal
functions.
Orthogonal Functions
For vectors,
𝑎⊥𝑏
means
𝑎 ⋅ 𝑏 = 0.
Similarly, two functions are orthogonal on [𝑎, 𝑏] if
⟨𝑓, 𝑔⟩ = 0
or equivalently,
𝑏
∫ 𝑓 (𝑥)𝑔(𝑥) 𝑑𝑥 = 0
𝑎
TERM 3
,FOUNDATIONS OF FOURIER ANALYSIS
The main idea we are building towards is:
A complicated function can be represented using simpler basis functions.
Eventually, those basis functions will be sines and cosines, giving us a Fourier series.
We already know how to break a vector into components along different directions.
Fourier analysis essentially does the same thing with functions.
The Vector–Function Analogy
Consider a vector
𝑎1
𝑎
𝑎 = [ 2 ].
𝑎3
A vector is essentially a collection of components.
Now imagine sampling a function 𝑓(𝑥) at discrete points:
𝑥 = 0, ℎ, 2ℎ, 3ℎ, …
We could represent the function values as
𝑓(0)
𝑓(ℎ)
𝑓 = 𝑓(2ℎ) .
𝑓(3ℎ)
[ ⋮ ]
As ℎ → 0, we sample the function at more and more points. In the limit, the function can
be thought of as an infinite-dimensional vector.
,This is important because if functions behave like vectors, we can borrow vector
concepts such as:
• dot products;
• length/norm;
• orthogonality;
• projections;
• basis vectors.
These become:
• inner products;
• function norms;
• orthogonal functions;
• projection onto basis functions;
• basis functions.
And projection onto basis functions is exactly what eventually gives us Fourier
coefficients.
Dot Product → Inner Product
For vectors,
𝑛
𝑎 ⋅ 𝑏 = ∑ 𝑎𝑗 𝑏𝑗 .
𝑗=1
The dot product tells us how much two vectors point in the same direction.
The function equivalent is the inner product.
For two real functions 𝑓(𝑥)and 𝑔(𝑥)on [𝑎, 𝑏],
𝑏
⟨𝑓, 𝑔⟩ = ∫ 𝑓 (𝑥)𝑔(𝑥) 𝑑𝑥
𝑎
Vectors Functions
𝒂⋅𝒃 ⟨𝑓, 𝑔⟩
sum components integrate functions
measures overlap/alignment measures overlap between functions
Function Norm
For a vector, its length is
, ∥ 𝑎 ∥= √𝑎 ⋅ 𝑎.
We define the length or norm of a function in exactly the same way:
∥ 𝑓 ∥= √⟨𝑓, 𝑓⟩
Therefore,
𝑏
∥ 𝑓 ∥= √∫ [ 𝑓(𝑥)]2 𝑑𝑥
𝑎
for real functions.
Normalised Functions
For a vector,
𝑎
𝑢𝑎 =
∥𝑎∥
creates a unit vector (its length is now one)
Similarly, a function can be normalised:
𝜙(𝑥)
𝜙norm (𝑥) =
∥𝜙∥
so that
∥ 𝜙norm ∥= 1.
This distinction becomes important when we talk about orthogonal vs orthonormal
functions.
Orthogonal Functions
For vectors,
𝑎⊥𝑏
means
𝑎 ⋅ 𝑏 = 0.
Similarly, two functions are orthogonal on [𝑎, 𝑏] if
⟨𝑓, 𝑔⟩ = 0
or equivalently,
𝑏
∫ 𝑓 (𝑥)𝑔(𝑥) 𝑑𝑥 = 0
𝑎
