Discontinuous Fourier transforms?

Question

What's an example (or even better a large class of examples) of an $L^2$ function whose Fourier transform is discontinuous?

Try $x \mapsto \mathbb{sinc}(x)$. – copper.hat May 11 '12 at 06:25 — copper.hat, May 11 '12 at 06:25
The answers and comments are sinc-ronizing. – copper.hat May 11 '12 at 06:57 — copper.hat, May 11 '12 at 06:57

score 3 · Accepted Answer · answered May 11 '12 at 06:00

3

Just take the inverse Fourier transform of your favourite discontinuous $L^2$ function.

answered May 11 '12 at 06:00

Robert Israel

470,583

for example, this is not hard to compute for a characteristic function – May 11 '12 at 06:42

score 2 · Answer 2 · edited Apr 13 '17 at 12:20

2

Robert Israel gives the most general answer, but here is an explicit example.

By scaling this answer, it is shown that the Fourier Transform of the $\mathrm{sinc}$ function $$ \mathrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x} $$ is the square bump function $$ \frac{\mathrm{sgn}(1+2x)+\mathrm{sgn}(1-2x)}{2} $$

edited Apr 13 '17 at 12:20

Community

1

answered May 11 '12 at 06:27

robjohn

353,833

Discontinuous Fourier transforms?

2 Answers2