Timeline for Group theoretical properties and symmetry based representation of common functions
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 31, 2011 at 11:31 | comment | added | Jeffrey | Yes, for a symmetry prescription to hold, the properties implied by it should hold (derivative exist, etc.). For simplicity, one may assume here that these are smooth, analytic functions. As for the issues on sym2: A function defines it's derivatives and antiderivatives (if they exist). Although not uniquely, but in some sense such a differential symmetry like sym2 maps the function to its self. Sym2 may be viewed as an act of three group elements: a reflection and a translation on the domain and an act of a group element, which represents a stepping on the derivative hierarchy. | |
| Jul 31, 2011 at 1:40 | comment | added | David Roberts♦ | 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? | |
| Jul 30, 2011 at 21:40 | comment | added | Pierre | oops, "Fourier coefficientS", plural! | |
| Jul 30, 2011 at 21:40 | comment | added | Pierre | 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). | |
| Jul 30, 2011 at 21:18 | history | edited | Jeffrey | CC BY-SA 3.0 |
added 867 characters in body; added 1 characters in body; deleted 1 characters in body
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| Jul 30, 2011 at 20:39 | comment | added | Jeffrey | By function I mean a map from the domain to the codomain. And yes, I forgot to add, for simplicity let's assume functions $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$. | |
| Jul 30, 2011 at 20:23 | comment | added | Yemon Choi | Functions with what domain and what codomain? Are you tacitly assuming both to be ${\mathbb R}$? | |
| Jul 30, 2011 at 20:18 | comment | added | Qiaochu Yuan | What do you mean by "function"? Why should you expect a generic function to have any symmetries in any reasonable sense? | |
| Jul 30, 2011 at 20:03 | history | asked | Jeffrey | CC BY-SA 3.0 |