In functional spaces, you try to decompose a function on an infinite base. Hilbertian geometry looks at functional spaces as vector spaces, in which a function f is a vector, so imagine you have an orthogonal basis (infinite) e(i) for i=1,..., n, ..., and a scalar product of vectors v and w, then you have f = sum ( . e(i) ) for i=1...n... infinity

Fourier sums define such specific functional spaces with base vector e defined by exponential of (i.omega.t).

There are many properties, for instance the Foruier transform of the derivative of a function, etc.

There are also a lot of physics where Fourier transforms are observed. For instance in diffraction of light by a thin slot...

BACK TO YOUR QUESTION

What do you mean in your question by "paired"?

Do you mean inverse Fourier transform of the Fourier transform?

Are you asking about invariants?

You may also be interested by the distribution theory (Laurent Schwartz), where the space observed is the set of linear forms on a functional space.

D is called a distribution if it is a linear function from the functional space C(infinite) of function indefinitely differentiable, to the set of real numbers R.

A function f gets mapped to by D, which is linear in f, and the result is a real number .

Maybe I misunderstood your interest. Can you explain in more details what you are looking for?

As examples for Table of Fourier Transform Pairs , you may follow the following

https://www.ethz.ch/content/dam/ethz/special-interest/baug/ibk/structural-mechanics-dam/education/identmeth/fourier.pdf

https://www.astro.umd.edu/~lgm/ASTR410/ft_ref1.pdf

http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_24/24_3_special_fourier_trnfm_pairs.pdf

Fourier analysis began by representing periodic functions in the form of a series called Fourier series, however

The analysis included other functions and became an important tool in mathematics. And Joseph Fourier

A French mathematician laid the foundations for mathematical theory of thermal proliferation, in his book Théorie analyticique de la chaleur

Who bore the name, is one of the most important books published in the nineteenth century

. Fourier's Fourier Series is an important tool in the approximation of functions, the appropriateness and analysis of experimental data, and has occupied an important position in many areas of mathematics Such as equations

Differential, integrative and analytical functions.

And then evolved to find Fourier series that had arisen in order to solve some mathematical physics issues such as thermal diffusion, but their effect in mathematical analysis is equally important. Some examples are sufficient to illustrate this idea. In the eighteenth century the idea of ​​the function itself was subject to argument, and the agreement was to name a function of all that could be expressed by

Such as polynomials, correct sequences and trigonometric sequences. But it was no longer an affiliate if the curve of the graph had a polygon shape. So many were surprised when Fourier series theory proved that many of these polygonal lines could be represented in triangular sequences, and Dirichlet was the first to put

The precise proofs of this matter were made in 1837. In 1861 Firestrass introduced the first example of a real function

Is continuous and does not accept the derivation at any point of this function was a Fourier series. We find another important example of the effect that the Twistorian series has left in the development of mathematics, because of its close connection to the concept of integration that suggests

His relationships. When Cantor wanted to solve the issue of unitary representation by Fourier series, he found within a theoretical framework

Riemann's integrals have different sets of numbers that have no bearing on this singularity. The study of these groups led them to lay the foundations of group theory, which was one of the most important discoveries of the nineteenth century. The theory of Lobig's integration was then founded on the work of Cantor, and in turn he gave the integration of Lobig, another aspect of the theory of series

Fourier and put it in a wonderful and integrated frame. Fourier series are used to represent periodic functions. In case of non-periodic function, Fourier integration is used and there are many sources in this direction