For vector \(v\) in the span of orthogonal basis \(v_1, ..v_{n}\):

\begin{equation} v = c_1 v_1 + \dots + c_{n} v_{n} \end{equation}

we can write:

\begin{equation} c_{j} = \frac{v \cdot v_{j}}{ v_{j} \cdot v_{j}} \end{equation}

Proof:

\begin{equation} \langle v, v_{j} \rangle = c_{n} \langle v_{1}, v_{j} \rangle \dots \end{equation}

which is \(0\) for all cases that’s not \(\langle v_{j}, v_{j} \rangle\) as the \(v\) are orthogonal, and \(\mid v_{j} \mid^{2}\) for the case where it is.

Fourier Series and how to find them.

For a function given at some interval of length \(l\), then the function can be written at:

\begin{equation} f(x) = \sum_{k=1}^{\infty} a_{k} \sin \qty( \frac{k\pi x}{l}) \end{equation}

or

\begin{equation} f(x) = \sum_{k=1}^{\infty} b_{k} \cos \qty( \frac{k\pi x}{l}) \end{equation}

Recall that because sin and cos are even and odd parts, the functions above force an even and oddness to your expansions. They will be particularly helpful for Dirichlet Conditions and Neumann Conditions.

requirements

Consider a function that has no periodicity, but that:

\begin{equation} f(x), -\infty < x < \infty \end{equation}

And assume that:

\begin{equation} \int_{\infty}^{\infty} |f(x)| \dd{x}, < \infty \end{equation}

important: look up! the integral of \(f(x)\) has to converge AND this means that the \(f(x)\) goes to \(0\) actually at boundaries.

(meaning the function decays as you go towards the end)

definition

a Fourier transform is an invertible transformation:

\begin{equation} f(x) \to \hat{f}(\lambda) \end{equation}

FDR is an American president.

• FDR and Teddy Roosevelt is
• Got Polio, which played in his favor => press agree to not photograph him when he was in a wheelchair
• Created the New Deal

Models himself after his cousin Teddy Roosevelt, and believed that charisma and moral leadership work. “Above all, try something… let the court shoot it if need to.”

He was able to gain single party control, wh.

Created Fireside Chats.

His wife, Eleanor Roosevelt, was very controversial.