Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Question

Initially I asked this question on mathoverflow. I however thought physicists may face this sort of problem more than mathematicians (I am an engineer). Due to that, I decided to ask here as well.

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier transform as follows:

\begin{equation} f(d/dx) g(x)=\int_{-\infty}^{\infty} f(i\omega)\mathcal{G}(\omega)e^{i\omega x}dw, \end{equation}

where $\mathcal{G}$ is the Fourier transform of $g(x)$. Say for instance, if $f(x)=x$, the above integral leads to $\frac{d}{dx}g(x)$ as expected. I am however stuck in calculating $f(i\omega)$ when $f$ is known numerically.

Question: How to evaluate $f(i\omega)$ given $N$ available samples?

Edit: Is this the right approach to compute $f(d/dx)$ numerically?

I am unsure about the selected tags below. Please help to add proper tags as well.

Answer 1

Very crudely, I'd suggest the following. Let's represent the numerical function as $$ f(x) = \sum_n f_n \delta(x-n\Delta x) , $$ where $f_n$ represents the numerical value. Then we represent the Dirac deltas as $$ \delta(x)=\int \exp(i2\pi xa)\ da $$ so that $$ f(x) = \sum_n f_n \int \exp(i2\pi xa-i2\pi na\Delta x)\ da . $$ It then means that $$ f(i\omega) = \sum_n f_n \int \exp(-2\pi a\omega-i2\pi na\Delta x)\ da . $$ It may be possible to solve the required integrals, assuming they are convergent.