Lecture 15 (Mar 25)

Delta function

$$
\mathcal{F}(1)=\delta(y)
$$

$$
\mathcal{F}(\delta)=1
$$

$$
\mathcal{F}(\cos 2\pi x) = \frac{1}{2} [ \delta(y-1)+\delta(y+1)]
$$

Some properties of Fourier Transform

1)
$$

\mathcal{F}(f(ax+b))=\frac{1}{a} e^{\frac{2\pi i by}{a}} \hat{f} \left( \frac{y}{a} \right)
$$
2)
$$

\mathcal{F}(f')=2\pi i y \mathcal{F}(f)
$$
3) Plancherel Identity
$$
\int |f(x)|^2 dx = \int |\hat{f}(y)|^2 dy
$$
4) $\hat{f}(0)=\int f(x) dx=$ are area under the function.

Convolution

$$
f*g(x)=\int_{-\infty}^{\infty} f(x-y) g(y) dy
$$

Properties:
1) Commutative
2) Linear

Convolution Theorem

$\mathcal{F}(f*g)(s)= \hat{f}(s) \cdot \hat{g}(s)$.

Reading Materials

Lecture Notes: p. 82-85, 88-90