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What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both sides).

For example, it is true if $f(x)=q^x$, for any $0<q<1$. This is also true for a Gaussian function.

What would be the general conditions for this to hold?

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