Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|cite|improve this question
Try $x \mapsto \mathbb{sinc}(x)$. –  copper.hat May 11 '12 at 6:25
The answers and comments are sinc-ronizing. –  copper.hat May 11 '12 at 6:57

2 Answers 2

up vote 3 down vote accepted

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

share|cite|improve this answer
for example, this is not hard to compute for a characteristic function –  user12014 May 11 '12 at 6:42

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 $$ \frac{\mathrm{sgn}(1+2x)+\mathrm{sgn}(1-2x)}{2} $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.