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Conjugate variables and the Fourier transform are often used to analyze different states of a single object. For example in Quantum Mechanics it can be used to describe changing information about position and momentum of a particle

But imagine that instead of different states you have different objects (with fixed information about each). I'm interested in knowing what properties a space/a set of those objects have. (and what generalisations of such spaces exist)

For example let's say you have a bunch of objects with outputs [a] and [b] (properties/variables [a] and [b]). When you consider all those objects together you see that [a] and [b] are functions related through the Fourier Transform.

Concerete example: one of such objects can be a box that always makes a single "click" sound (defined time and undefined frequency)... and the other object can be a box that always plays a single note (undefined time and defined frequency)... and et. cetera

What is the space/set of those boxes? What properties does it have?

https://en.wikipedia.org/wiki/Integral_transform

an integral transform maps a function from its original function space into another function space via integration

I want to know what is the function space in my example above... and what properties does it have!

Why I'm asking. Imagine you have a bunch of seemingly separate objects and have a suspicion that they are all related through the Fourier Transform. To make sure if it is the case you would like to know more about the space of those objects.

The outputs can be probabilities as they are in the Quantum Mechanics case (hence relatedness to statistics) and the problem can theoretically arise in machine learning or rather in some data-mining/analysis endeavour in general

P.S.: I apologize for being a layman and maybe unable to appreciate all the information you could give... but believe me it's important for me!

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