1
$\begingroup$

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, and was likely far beyond the realm of my ability anyway.

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

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 $m$ 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 $$[a,b]=[m-d,m+d].$$

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 (real part of the) Fourier transform would be the first positive zero, and the first negative 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?

$\endgroup$
3
  • $\begingroup$ the Fourier cosine transform of $1/(x^2+1)$ is $e^{-|k|}$ -- it does not seem to satisfy your conjecture about "infinitely many isosceles intervals" (it doesn't have any zeroes at all). $\endgroup$ Dec 4, 2020 at 20:46
  • $\begingroup$ Thank you. I suppose I could reformulate the question to where zeros exist, although that might be delaying the inevitable. $\endgroup$ Dec 4, 2020 at 22:07
  • $\begingroup$ a Fourier transform could have only a finite number of zero's, so I really don't see why the presence of "infinitely many isosceles intervals" would be universal. $\endgroup$ Dec 4, 2020 at 23:00

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.