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I am working on the proof of the Fourier Transform of the derivative of a function. I am accompanied by some proof lines but having some issue in one of the integral evaluation. I searched out its proof in this forum, and I found out couple of proof, but there are some ambiguities in that which I want to clear.

I have defined the Fourier Transform by the following equation:

$\mathfrak{F}\{f(t)\} = \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t}} dt$

where f is a vector fucntion. i.e., $f(R^1\rightarrow R^2)$

and I want to verify the Fourier of the derivative of function $f$. i.e.,

$\mathfrak{F}\{f'(t)\} = j\omega \mathfrak{F}\{f(t)\}$

Proof:

Step.1: I took the left hand side of the property and apply the Fourier Transform defintion to obtain:

\begin{equation} \mathfrak{F}\{f'(t)\} = \int_{-\infty}^{+\infty} {f'(t)e^{-j \omega t}} dt \end{equation}

Step.2: I applied the integration by parts to obtain the following equation:

\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} - \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t} {(- j \omega)}} dt \end{equation}

or

\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} + {j \omega} \int_{-\infty}^{+\infty} {f(t)e^{-j \omega t}} dt \end{equation}

or

\begin{equation} \mathfrak{F}\{f'(t)\} = {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} + j\omega \mathfrak{F}\{f(t)\} \end{equation}

This last equation is almost same as given in one of the answer to a question on the following link.

Fourier Transform Properties - Proving

Now to prove the derivative property the first term of the last equation should be equal to zero. i.e.,

\begin{equation} {e^{-j \omega t}f(t)}\mid_{-\infty}^{+\infty} = 0 \end{equation}

In another answer to one of the question regarding derivative property in this forum, it says "The first term must vanish, as we assume $f$ is absolutely integrable on $R$" which can be found on the following link:

Fourier Transform of Derivative

By the absolute integrability, it means that integral of norm of function would be convergent, but:

How would I prove it (i.e., the first term equal to zero).

I will highly appreciate the response to the question.

Thanks in advance and best regards.

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