Band-limited signal $f(x)$ and its absolute value $|f(x)|$

Question

Suppose a function $f(x)$ in the spatial domain is band-limited with band limit $u_{max}$ on the frequency domain. Then is the absolute value $|f(x)|$ also band-limited? I don't think this is true. Can someone give me a counterexample?

Answer 1

The absolute value of a Fourier transformable function does not always give a Fourier transformable function. In the traditional theory the Fourier Transform is defined for infinitely differentiable functions (the Schwarz class). What happens to the derivatives at points where $f(x_k)=0$ is in general that we can not guarantee continuity so the absolute value of the function won't be transformable.

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Note Below is a partial answer to another question how to find a sufficient condition for smoothing of functions absolute values to make them Fourier transformable and band limited.

A band limited real-valued function can be written in the fourier domain as $$F(w)=h_{-w,w}(w)G(w)$$ where $G(w)$ is a function having the property $\overline{G(w)}=-G(-w)$, and where $$h_{a,b}(w) = \begin{cases}{0, w < a\\1, a < w < b\\0, b < w}\end{cases}$$ encodes the frequency band by multiplying with 1 inside and 0 outside.

Now the squaring (multiplication) in the time domain is the self-convolution $(F*F)(w)$. Now with the help of $h$ maybe you can investigate this convolution.