Example Problem Set 4

Suppose that some continuous function f(x) has Fourier Transform $F(\mu)$. Now consider the consequences in the Fourier domain of each of the following operations upon f(x):

1.

What will be the Fourier Transform of the m^th derivative of f(x) with respect to x: ${\displaystyle \frac{d^mf(x)}{dx^m}}$?

2.

What will be the Fourier Transform after shifting f(x) by a distance $\alpha$: $f(x-\alpha)$?

3.

What will be the Fourier Transform after dilating f(x) by a factor $\alpha$: $f(x/\alpha)$?

4.

Suppose that f(x) is convolved with another function g(x)whose Fourier Transform is $G(\mu)$. What will be the Fourier Transform $H(\mu)$ of the convolution result h(x) where

$$h(x)=f(x)*g(x)=\int_{-\infty}^{+ \infty} f(\alpha)g(x-\alpha) d\alpha ?$$

5.

Suppose now that a two-dimensional continuous function f(x,y) has a 2D Fourier Transform $F(\mu,\nu)$. Define the Laplacian operator $\nabla^{2}$and express the 2D Fourier Transform of $\nabla^{2}f(x,y)$.