Condition for existence of Fourier transform?

Question

We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits.

So I want to ask

is there any condition for existence of Fourier transform ? What is the limit for Fourier transform to converge?

Answer 1

This is the definition: $$\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx,$$ for any real number $ξ$.

We assume $f(x)$ is an integrable function, Lebesgue-measurable on the real line, and satisfy: $$\int_{-\infty}^\infty |f(x)| \, dx < \infty.$$