## Fourier Series ##
### Classical Fourier Series
### Classical Fourier Series
$$
f(x) =\sum\_{n = 0}^{+\infty} B\_n \sin (\frac{nx \pi}{l})
$$
$$
g(x) = \frac{1}{2} A\_0 + \sum\_{n = 0}^{+\infty} A\_n \cos (\frac{nx \pi}{l})
$$
$$
h(x) = \sum\_{n = -\infty}^{+\infty} C\_n e^{i \frac{nx \pi}{l}}
$$
### Fourier-Sine Series
$$
f(x) =\sum\_{n = 0}^{+\infty} B\_n \sin (\frac{nx \pi}{l})
$$
* Naturally defines a function on $x \in [0,l].$
* Extends as an ODD function to $x \in [-l,l].$
### Coefficient Formula
$$
B\_n = \frac{1}{l} \int\_0^l f(x) \sin (\frac{nx \pi}{l}) dx.
$$
### Fourier-Cosine Series
$$
g(x) = \frac{1}{2} A\_0 + \sum\_{n = 0}^{+\infty} A\_n \cos (\frac{nx \pi}{l})
$$
* Naturally defines a function on $ x \in [0,l]$.
* Extends as EVEN function to $x \in [-l,l].$
### Coefficient Formula
$$
A\_n = \frac{1}{l} \int\_0^l f(x) \cos (\frac{nx \pi}{l}) dx.
$$
### Full Fourier Series
$$
h(x) = \sum\_{n = -\infty}^{+\infty} C\_n e^{i \frac{nx \pi}{l}}
$$
* Naturally defines a function on $x \in [-l,l].$
* Not necessarily even, odd or ${\mathbb{R}}$-valued.
* Infinite summation goes in both directions.
### Coefficient Formula
$$
C\_n = \frac{1}{2l} \int\_{-l}^l e^{-i \frac{nx \pi}{l}} f(x) dx.
$$
### Orthogonality
* Linear Algebra Point-of-View
* Coefficient Formula viewed as inner products
* Sines, cosines, exponentials viewed as unit vectors
$$
\langle e^{imx \frac{\pi}{l}}, e^{inx \frac{\pi}{l}} \rangle = \int\_{-l}^l e^{imx \frac{\pi}{l}} {\overline{e^{inx \frac{\pi}{l}}}} \frac{dx}{2l}
$$
$$
= \begin{cases}
1 ~{\mbox{if}}~ m = n \\\\
0 ~{\mbox{if}}~ m \neq n. \\\\
\end{cases}
$$
* Similar property for sines and cosines.
### Convolution
The [convolution](http://en.wikipedia.org/wiki/Convolution) between two functions $f,g$ is given by the formula
$$
(f*g)(x) = \int f(x-y)g(y)dy.
$$
### Convolution and Fourier Coefficients
### Oscillation Decay Problem
How does the integral
$$
J_\lambda = \int \sin(\lambda x) \phi(x)dx
$$
behave as $\lambda \rightarrow \infty$?
(Assume $\phi$ is nice, compactly supported, ....)
### Intuition: Cheese Grater
### Intuition: Cheese Grater
### Intuition: Cheese Grater
### Intuition: Cheese Grater
### Intuition: Cheese Grater
### Intuition: Cheese Grater
### Integration by Parts
$$
J_\lambda = \int \sin(\lambda x) \phi(x)dx?
$$
Substitute
$$
\sin (\lambda x) = - \frac{1}{\lambda} \frac{d}{dx} \cos (\lambda x)
$$
then integrate by parts. This reveals the decay.
### Decay via Orthogonality
### (Non)Stationary Phase Method
How does the integral
$$
I_\lambda = \int \sin(\lambda x^2) \phi(x)dx
$$
behave as $\lambda \rightarrow \infty$?
(Assume $\phi$ is nice, compactly supported, ....)
### (Non)Stationary Phase Method
### (Non)Stationary Phase Method
### (Non)Stationary Phase Method
### (Non)Stationary Phase Method
### Van Der Corput Lemma
***
For a nice function $\phi$, the oscillatory integral decays:
$$| \int \sin(\lambda x^2) \phi(x)dx ~ | \leq C_\phi \frac{1}{\sqrt{\lambda}}$$
***
### Proof of Van Der Corput Lemma
* Suppose $\phi$ is supported on $[-a, a]$.
* Chop the integral into three pieces:
* $[-a, -\delta]$
* $[-\delta, \delta]$
* $[\delta, a]$
* Integrate by parts on outer intervals.
* Crudely estimate inner integral.
* Optimize over $\delta$.
### Exercise: Generalized Decay Estimate
***
For a nice function $\phi$, the oscillatory integral decays:
$$| \int \sin(\lambda x^k) \phi(x)dx ~ | \leq C_\phi \lambda^{-\frac{1}{k}}$$
***
### Application: Dispersion
Consider the free Schrödinger equation:
$$ \begin{cases}
i \partial\_t u = \partial_x^2 u \\\\
u(0, x) = u\_0(x)
\end{cases} $$
The solution $u: [-T, T] \times {\mathbb{R}} \longmapsto {\mathbb{C}}$ is called a *free Schrödinger wave* and arises in quantum mechanics.
### Convergence Notions
**Pointwise Convergence:**
**Uniform Convergence:**
**$L^2$ Convergence:**
### Ubiquity
* Expansions in Eigenfunctions of the Derivative
* Derivatives act like diagonal matrices
* differential equations $\rightarrow$ algrebraic equations.