# Group theoretical properties and symmetry based representation of common functions

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.

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What do you mean by "function"? Why should you expect a generic function to have any symmetries in any reasonable sense? – Qiaochu Yuan Jul 30 '11 at 20:18
Functions with what domain and what codomain? Are you tacitly assuming both to be ${\mathbb R}$? – Yemon Choi Jul 30 '11 at 20:23
By function I mean a map from the domain to the codomain. And yes, I forgot to add, for simplicity let's assume functions $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$. – Jeffrey Jul 30 '11 at 20:39
this may not be what you're looking for, but Fourier analysis says that functions with the symmetry $f(x)= f(x + 2\pi)$ are best represented by their Fourier coefficient. More generally, conjugation-invariant functions on a Lie group can be approximated by characters of representations (Peter-Weyl theorem). – Pierre Jul 30 '11 at 21:40
Your 'symmetry' 2 is very different from symmetry 1, as the first is an invariance under a group acting on the domain, whereas the second is a differential equation. In questions like this you need to specify what space of functions you are considering. Smooth? Analytic? – David Roberts Jul 31 '11 at 1:40