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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 !

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    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 :)2012-10-17

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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 $k$.

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    Depends on who's doing the counting.2012-10-18