# Fourier Series

We can write any periodic function over a period $T$ as $f(t)=a_{0}+∑_{k=1}a_{k}cos(T2πkt )+∑_{k=1}b_{k}sin(T2πkt )$

- If $T=2π$, it simplifies the expression

with coefficients are given by solving the following integrals: $a_{0}=2π∫_{0}f(t)(dt) $ $a_{k}=∫_{0}cos_{2}(kt)dt∫_{0}f(t)cos(kt)dt $ $b_{k}=∫_{0}sin_{2}(kt)dt∫_{0}f(t)sin(kt)dt $

Now, given the above sinusoidal expression of a function $f(t)$ (with period $2π$) $f(t)=a_{0}+∑_{k=1}a_{k}cos(kt)+∑_{k=1}b_{k}sin(kt)$

we can express it more concisely using Euler’s Formula as $f(t)=∑_{k=−∞}c_{k}e_{ikt}$ where the $c_{k}$ coefficients are complex numbers.

But can you express

`c_k`

as a sum of`a_k`

and`b_k`

?Yup.

### Approximation

An approximation of a function could be achieved by truncating the series to a finite number of sinusoids: $f(t)≈∑_{k=−M}c_{k}e_{ikt}$

For $N$ points (assume $N$ is even), we will use $N$ degrees of freedom (i.e., $N$ coefficients) to exactly interpolate the data.

$f(t)≈∑_{k=−2N+1}c_{k}e_{T(2πi)kt}$

Plugging in each of our $N$ data points $(t_{n},f_{n})$ into the expression will give us $N$ equations, involving unknowns coefficients, $c_{k}$. This leads us to the Discrete Fourier Transform!

### From MATH213

Definition 1:

`L^2`

functionsA complex valued function $f$ is in the class $L_{2}([a,b]])$ if

$∫_{a}∣f(x)∣_{2}dx$ exists and is finite.$f$ is in the class $L_{2}$ if $∫_{−∞}∣f(x)∣_{2}dx$

exists and is finite.

We want to write $f(t)=∑_{n=−∞}c_{n}e_{τ2πnjt}$ for $t∈[−τ/2,τ/2]$

Our goal is to find $c_{n}$.

Definition 3: Fourier Series - Complex Form

If $f∈L_{2}([−τ/2,τ/2])$ then the Fourier series in complex form of $f(t)$ is $∑_{n=−∞}c_{n}e_{τ2πnjt}$ where the $c_{n}$ are found by projecting $f$ into the basis of complex exponentials.

Theorem 3: Fourier Coefficients for Series in Complex Form

If $f∈L_{2}([−τ/2,τ/2])$ then the Fourier coefficients $c_{n}$ of $f(t)$ are

$c_{n}=<f(t),e_{τ2πnjt}>$ $c_{n}=τ1 ∫_{2τ}f(t)e_{−τ2πnjt}dt$

If $f$ is real valued than $c_{n}=c_{−n} $.

- I was so confused for a second, but $<a,b>$ is the Inner Product

Theorem

For a real valued, $τ$ periodic function $f∈L_{2}([−τ/2,τ/2])$:

- If $f$ is an even then the Fourier series can be simplified to a sum of $cos$ waves.
- If $f$ is an odd then the Fourier series can be simplified to a sum of $sin$ waves.

Theorem 2: Fourier Sine and Cosine Coefficients

If $f$ is a real valued function that is in $L_{2}([−τ/2,τ/2])$ then

If $f$ is even then the Fourier cosine series for $f$ is $∑_{n=0}c_{n}cos(τ2πn t)$ where $c_{n}=f(x)={⟨f(t),1⟩2⟨f(t),cos(τ2πn t)⟩ n=0n>0 $

If $f$ is odd then the Fourier sine series for $f$ is $∑_{n=1}s_{n}sin(τ2πn t)$

where $s_{n}=2⟨f(t),sin(τ2πn t)⟩$

- It’s a little hard to memorize this. Instead, memorize theorem 3 and derive it

### Convergence of Fourier Series

See PWC1.

Theorem 5: Convergence of Fourier series

Let $f_{p}$ be the Periodic Extension of a function $f∈L_{2}([−τ/2,τ/2])$.

- The Fourier series of $f$ converges in the $L_{2}$ norm (also in the mean and almost everywhere) to $f$ (and also $f_{p}$) on any finite subinterval of $[−τ/2,τ/2]$.
- If$f_{p}$ is piecewise $C_{1}$ then the Fourier series of $f$ converges pointwise to $f_{p}$ for all $x∈R$
- If $f_{p}$ is piecewise $C_{1}$ and continuous then the Fourier series of $f$ converges uniformly $f_{p}$ on any finite interval of $R$.