Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Question

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ?

I am not necessarily looking for a closed formula.

Even some iterative method would be fine.

Thanks.

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EDIT:

I am doing an actual computation and trying to do most of the calculations in the frequency domain.

basically I have a point wise operation on $f(x,y)$, that taking the maximum between the value of the function and 0 $\rightarrow max(0,f(x,y))$ , this operation is non linear.

however I only have $\mathscr{F}\{f(x,y)\}$ and would like to avoid doing the inverse transofrm.

I know that I can calculate for example, $\mathscr{F}\{\frac{\partial}{\partial x} f(x,y)\} = \mathscr{F}\{\frac{\partial}{\partial x}\} \circ \mathscr{F}\{ f(x,y)\}$

where $\circ $ is point wise multiplication

and $\mathscr{F}\{\frac{\partial}{\partial x}\}$ is the fourier transform of the corrosponding convolution kernel - which is constant for any $f(x,y)$.

in my case the operation is not linear and therefore cannot be described with a convolution... I would like to know if it is possible to aproximate it using somthing like:

$\mathscr{F}\{max(0,f(x,y))\} \approx a_1 \mathscr{F}\{K_1\} \circ \mathscr{F}\{f(x,y)\} +a_2 \mathscr{F}\{K_2\} \circ \mathscr{F}\{f(x,y)\} + \dots + a_n \mathscr{F}\{K_n\} \circ \mathscr{F}\{f(x,y)\}$

btw if it helps, I have found a post that gives a good approximation for $ max(0,x)$

Answer 1

Just a couple of observations (I will deal with functions of only one real variable).

If you have a smooth approximation to $max(0,x)$, say $g(x)\simeq max(0,x)$, then you have a smooth approximation to $max(0,f(x))$. Now,

$$ \frac{\partial g}{\partial x}=\frac{\partial g}{\partial f}\frac{\partial f}{\partial x} $$

The Fourier transform of $\partial g/\partial x$ is then the convolution between the Fourier transform of $\partial g/\partial f$, which you have - but you do not want - to calculate, and that of $\partial f/\partial x$, which you already have.

Depending on the values of $f$, if you expand the function $g(x)$ in a Taylor series, then you will have polynomial approximation with powers of $f(x)$, so you will end up calculating convolutions, with not much computational benefit.