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### Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. ...
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### Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
What is the difference between Fourier integral and Fourier transform? I know that for Fourier integral, the function must satisfy $\int_{-\infty}^\infty |f(t)| dt < \infty$, but what if I have a ...