## Fourier Series ## ![utlogo](http://www.math.toronto.edu/colliand/images/utlogo.jpeg)
### 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
### 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
### Convolution The [convolution](http://en.wikipedia.org/wiki/Convolution) between two functions $f,g$ is given by the formula $$(f*g)(x) = \int f(x-y)g(y)dy.$$ ![A Graphical Representation of Convolution](http://upload.wikimedia.org/wikipedia/commons/6/6a/Convolution_of_box_signal_with_itself2.gif) ![Second Graphical Representation of Convolution](http://upload.wikimedia.org/wikipedia/commons/b/b9/Convolution_of_spiky_function_with_box2.gif)
### Properties
### Convolution and Fourier Coefficients
### Oscillation/Decay
### 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 ![CheeseGrater1](./assets/sine.png)
### Intuition: Cheese Grater ![CheeseGrater2](./assets/sine2.png)
### Intuition: Cheese Grater ![CheeseGrater4](./assets/sine4.png)
### Intuition: Cheese Grater ![CheeseGrater8](./assets/sine8.png)
### Intuition: Cheese Grater ![CheeseGrater16](./assets/sine16.png)
### Intuition: Cheese Grater ![CheeseGrater32](./assets/sine32.png)
### 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 ![Square1](./assets/square1.png)
### (Non)Stationary Phase Method ![Square2](./assets/square2.png)
### (Non)Stationary Phase Method ![Square4](./assets/square4.png)
### (Non)Stationary Phase Method ![Square8](./assets/square8.png)
### 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
### Fourier Synthesis?
### Convergence Notions **Pointwise Convergence:** **Uniform Convergence:** **$L^2$ Convergence:**
### Convergence Theorems
### Ubiquity
### Ubiquity * Expansions in Eigenfunctions of the Derivative * Derivatives act like diagonal matrices * differential equations $\rightarrow$ algrebraic equations.