(Due next Wednesday)

We will use the Boas convention for Fourier transformation (or see Friday's note).

1. Discrete Fourier Transformation for $N=3$. Suppose $f(x)$ is given by $$ f(x) = \delta_{x,0} $$ where $\delta_{i,j} = 0$ is $i\neq j$ and $=1$ if $i=j$.

Find the Fourier transformation $F(p)$. (You discovered that 'peak function' in $x$ space is sent to 'planewave' in $p$ space. )

What function $f(x)$ will have Fourier transformation $F(p) = \delta_{p,0}$?

2. Recall that if $f(x) = 1/(1+x^2)$, then its Fourier transformation is $F(p) = (1/2) e^{-|p|}$. Can you verify Parseval's Equality in this case?

3. Let $f(x) = 1$ for $x \in [0,1]$. Compute the convolution $(f\star f)(x)$. Can you plot it? What's the Fourier transformation of $f$ and $f \star f$? (The one for $f$ is already done in HW6).