# Limits of a Fourier transform

Consider a function $f(t)$ satisfying the following properties $$\lim_{t\to\pm\infty} f(t) = f_0,~~~~~\lim_{t\to\pm\infty} f'(t) = 0 \tag1$$ Consider now the Fourier transform of this function $$g(\omega) = \int_{-\infty}^\infty dt \, e^{i \omega t} f(t) \implies f(t) = \int_{-\infty}^\infty \frac{d\omega}{2\pi} e^{- i \omega t} g(\omega)$$ How can I express the properties (1) of the function $f$ in terms of $g$??

If required, it can be assumed that $f(t) = f_0$ in some region $|t| > t_0$ for sufficiently large $t_0$.

PS - I understand that the function $f(t)$ is not integrable, and hence $g(\omega)$ might contain Dirac delta functions. That is fine.

I am not quite sure how to proceed with this, and any help will be appreciated. Thanks.

## 1 Answer

Look at $f - f_0$. I can extend your hypotheses and suppose that this is a nice function with well defined and nice Fourier transform. Hence the Fourier transform of $f$ will be a nice function plus $2\pi f_0$ times the Dirac delta distribution.

When I use the word "nice", it is meant to be vague. But your question was similarly vague.

• I don't mind the term "nice". Good enough for me for now. I will do as you said. Thanks. – Prahar Jan 6 '14 at 2:45