Dear All,

I'm searching for some references, whether someone already studied the group theoretical properties of functions. There are some very basic symmetries, like parity, but is there a set of well defined symmetry transformations that could lead to a non-arbitrary classification of functions? Is there a way to uniquely define functions by their symmetries?

I have the feeling that the commonly used representation to deal with functions, compositions, derivatives, integrals etc. is not the best one. Also, from the computational point of view, this representation has redundancies (ie. different combination of functions can lead to identical expressions) that leads to inconveniences in symbolic algebra systems.

So, if there are publications related to this, or an already accepted name of this field, let me know, so I will know what to search for.

Edit 1: Example problem: Let's try to define the function $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$, $f(x) = \sin(x)$ by using only it's symmetries like below:

Symmetry 1: $f(x) = f(x + 2\pi), \forall x$

Symmetry 2: $f'(x) = f(\frac{\pi}{2} - x), \forall x$

where prime denotes differentiation w.r.t. x. Of course there are lots of similar identities.

One may ask the following question: Is there a set of symmetries like above, such that only $f(x) = \sin(x)$ satisfies all of them? If the set is infinite, is there a systematic way to generate all necessary symmetry rules?

Note: to be systematic, in the rule definitions one may only use functions defined in the same way, that is, by symmetries. For simplicity let's assume that the basic arithmetic functions already defined. Same problems applies to the differentiation / integration.