Fourier Series

What is the formula for a fourier series

$$S\left(x\right)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_{n}\cos\left(nx\right)+\sum_{n=1}^{\infty}b_{n}\sin\left(nx\right)$$

How do we find the coefficient a₀

\frac{1}{\pi}\int_{-\pi}^{\pi}f\left(x\right)1\differentialD x

How do we find the coefficient a_m

a_{m}=\frac{1}{\pi}\int_{-\pi}^{\pi}f\left(x\right)\cos\left(mx\right)\differentialD x

How do we find the coefficient b_m

$$b_{m}=\frac{1}{\pi}\int_{-\pi}^{\pi}f\left(x\right)\cdot\sin\left(nx\right)$$

Definition of the inner product

\langle f,g\rangle=\frac{1}{\pi}\int_{-\pi}^{\pi}f\left(x\right)g\left(x\right)\differentialD x

Theorem for Fourier coefficients of even and odd functions

• if f(x) is even, $b_{n}=0$ for all $n\ge1$

• if f(x) is odd $a_{n}=0$ for all $n\ge1$, including$a_0$

Fourier convergence theorem

If $f:\left\lbrack-\pi,\pi\right\rbrack\to\mathbb{R}$ is piecewise continuously differentiable, then at $x\in\left\lbrack-\pi,\pi\right\rbrack$

$$\lim_{N\to\infty}S_{N}\left(x\right)=S\left(x\right)=\frac{f\left(x^{+}\right)+f\left(x^{-}\right)}{2}$$

In particular, if f is continuous at x, then S(x) = f(x).

Periodic extensions definition

Parseval’s theorem definition

If $f:\left\lbrack-\pi,\pi\right\rbrack\to\mathbb{R}$ is piecewise continuously differentiable with Fourier coefficients $a_{n},n\ge0$ , $b_{n},n\ge1$ , then

\|f\|^2=\frac{1}{\pi}\int_{-\pi}^{\pi}f^2\left(x\right)\differentialD x=\frac{a_0^2}{2}+\sum_{n=1}^{\infty}\left(a_{n^{}}^2+b_{n}^2\right)

Parseval’s for complex exponential series

?? (same application as normal Parseval’s, just using complex fourier series definition)

Inner product for complex-valued functions

$f,g:\left\lbrack-\pi,\pi\right\rbrack\to\mathbb{C}$ is\langle f,g\rangle=\frac{1}{\pi}f\left(x\right)\overline{g}\left(x\right)\differentialD x

where$\overline{g}(x)$ is the complex conjugate of g(x).

Complex Fourier series definition

S\left(x\right)=\sum_{n=-\infty}^{\infty}c_{n}e^{\imaginaryI nx},c_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f\left(x\right)e^{-inx}dx

What is the norm of a function f?

Let $f:\left\lbrack-\pi,\pi\right\rbrack\to\mathbb{R}$ such that

\|f\|=\sqrt{\langle f,f\rangle}=\left(\frac{1}{\pi}\int_{-\pi}^{\pi}f^2\left(x\right)\differentialD x\right)^{\frac12}