Mth3006 Lecture 3

• Fourier transform - four properties (emphasis on scaling and differentiation).
• Convolution.

How Do You Prove the Scaling Property of Fourier Transforms?

… to be done at some point

How Do You Prove the Differentiation Property of Fourier Transforms?

PROOF: $\mathcal{F}[f^{\prime }(t)]=i\omega \tilde{f}(\omega )$

$$\begin{array}{rl}\mathcal{F}[f^{\prime }(t)]& =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{f}^{\prime }(t)e^{i\omega t}dt\\ & =\frac{1}{\sqrt{2\pi }}[e^{-i\omega t}f(t){]}_{-\infty }^{\infty }+\frac{1}{2\pi }{\int }_{-\infty }^{\infty }i\omega e^{-i\omega t}f(t)dt\\ & =i\omega \tilde{f}(\omega )\end{array}$$

On the second step we have used the fact that $f(t)\to 0$ as $t\to \pm \infty$, which must be true for the Fourier transform to exist.

Applying again, we find that $\mathcal{F}[f^{\prime \prime }(t)]=i\omega \mathcal{F}[f^{\prime }(t)]=-{\omega }^2\tilde{f}(\omega )$.

How Do You Prove the Translation Property of Fourier Transforms?

… to be done at some point

How Do You Prove the Exponential Multiplication Property of Fourier Transforms?

… to be done at some point

What is De-Convolution?

In practice, we find the resolution function from the observed function to find the original function - “de-blurring” a measurement, or image. Maybe this is with machine learning or similar, though.

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