Take the tour ×  The set of $\mathcal{L}^p$ functions is closed
under addition, multiplication and convolution operations. I'd
like to know an example of a set of functions which is closed
under convolution and not under addition and multiplication.
share|improve this question The easiest example is the standard
orthonormal basis in $\ell^2(\mathbb{N})$, I presume. –  t.b.
Jul 27 '11 at 8:12 @Theo Buehler : I do not know of such a
thing, could you please give a link or explain. –  Rajesh D Jul
27 '11 at 8:28 @Rajesh: See
en.wikipedia.org/wiki/Sequence_space. –  joriki Jul 27 '11 at
8:37 Let $e_n$ be the sequence that is zero except at place $n$.
The convolution $c=(c_n) = a \ast b$ of two sequences $a=(a_n)$
and $b=(b_n)$ is given by $c_n = \sum_{k} a_k \cdot b_{n-k}$.
Using this formula you have $e_{m+n} = e_m \ast e_n$. But
obviously the set $\{e_m\} \subset \ell^p$ is neither closed
under addition nor multiplication. This convolution arises when
looking at the Fourier transform $\mathcal{F}(f\cdot g) = \m