# reference request – Fourier Transforms exhibiting symmetries about their critical points

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?)

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Let $$a be numbers such that $$f(a),f(b)=0,$$ and there does not exist a number $$c$$ such that $$a 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 $$a,b$$
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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.
So does the zero function.

In general, I call an interval (of a function) that satisfies these properties isosceles, from the triangle whose vertices are located at $$(a,0),(b,0)$$ and $$(c,f(c))$$.

The obvious example of this in the (complex part of the) Fourier transform would be the first positive real zero, and the first negative real zero. Since the maximum of the Fourier transform is at $$hat{f}(0)$$, and the Fourier transform (of a real function) is Hermitian, that interval is isosceles.

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 there are infinitely many intervals of the Fourier cosine transform that satisfy this property.

All that being said, here’s an explicit formulation of my questions.

I haven’t looked extensively for other functions with “isosceles intervals”. Would a paper on functions satisfying these properties be publishable?

Where can I find papers about this feature of the Fourier Transform, assuming they exist? If not, is this “infinitely many isosceles intervals” conjecture about the Fourier Transform within my reach?