Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be worth asking about. I also had an idea similar to this (Fourier transform of the critical line of zeta?) but sadly, it's already been done.

$%The following sentence is horrible to read. Note to self: Rewrite the sentence later.%$

Let $a<b$ be numbers such that $$f(a),f(b)=0,$$ and there does not exist a number $c$ such that $a<c<b$ and $$f(c)=0.$$ For clarity, I will define zeroes satisfying these properties as adjacent zeros.

Certain functions exhibit symmetries about their adjacent zeros. For example, it is trivially verified that the sine function has its critical points directly between their adjacent zeros i.e., given a minimum or maximum $c$ of the sine function, we have an interval of two adjacent zeros $a,b$ $%This next part of the sentence has two nested "such that"s. Fix later to improve readability.%$ such that there exists a real number $d$ such that $$[a,b]=[c-d,c+d].$$

Translating left or right obviously doesn't affect the truth of this statement. $%not rigorous, fix later%$ Therefore, the cosine function also satisfies these properties.

However, intuitively, just from looking at various Fourier transforms (where the Fourier transform doesn't have a Dirac delta function in its domain) it seems that both the real and complex part of the Fourier Transform exhibit these properties.

Where can I find papers about this property, or, if they don't exist, is this within the reach of me proving?