I just saw a problem asking for an example of an algebra of real functions on the interval $[-1,1]$, which do not contain non-zero polynomials and nonzero trigo functions.

I think I just caught one : all the rational functions $\frac{p(x)}{q(x)}$ such that $deg(p) < deg(q)$ and $q(x) \neq 0$ on the interval $[-1,1]$.

However, this one seems a bit weird, do you have any other examples ? Maybe an algebra of functions containing some exponentials, or some logs, or just functions which are not $C^{\infty}$... Any other example appreciated !

  • $\begingroup$ Do you want "algebras" without unit that do not contain any polynomial (including the constant function $1$)? Or do you allow algebras that do not contain any non-constant polynomial? $\endgroup$ – Stephen Oct 17 '12 at 22:39
  • $\begingroup$ Hey. Well, I don't know. It's a question on some old prelim exam in my University, and if you think of the algebra of constant functions, I think this answer would not be accepted. However, if you need them for another algebra of functions, go ahead and take it :) $\endgroup$ – Bogdan Oct 17 '12 at 23:18

Presumably, if nonzero constant polynomials are not allowed, this "algebra" is not assumed to include a unit.

Well, e.g. take the algebra generated by $\exp$, i.e. all linear combinations over $\mathbb C$ (or whichever subfield thereof you want to use for scalars) of the functions $\exp(kx)$ for positive integers $k$.

  • $\begingroup$ Haha, I was being dumb. I eliminate this example by the stupid argument "But $e^x$ + $e^x$ is not of this same form". It doesn't need to. So yeah, the algebra of functions generated by $exp(x)$ is good. Thanks, this satisfies my curiosity of finding a good example. $\endgroup$ – Bogdan Oct 18 '12 at 0:11
  • $\begingroup$ I thought that $e^x$ counted as a trig function. $\endgroup$ – Stephen Oct 18 '12 at 15:35
  • $\begingroup$ Depends on who's doing the counting. $\endgroup$ – Robert Israel Oct 18 '12 at 17:47

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