What's an example (or even better a large class of examples) of an $L^2$ function whose Fourier transform is discontinuous?
Discontinuous Fourier transforms?
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fourier-analysis
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2The answers and comments are sinc-ronizing. – 2012-05-11
2 Answers
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Just take the inverse Fourier transform of your favourite discontinuous $L^2$ function.
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0for example, this is not hard to compute for a characteristic function – 2012-05-11
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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} $