Hi,

can anyone please explain to me why convolution in $\displaystyle L^1 $ does not have an identity element?

i.e.

there is no $\displaystyle u \in L^1(R^n)$ such that

$\displaystyle u * g = g$ $\displaystyle \forall g \in L^1(R^n) $

Thanks

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- Aug 22nd 2008, 08:08 AMmcbain666Identity Element for Convolution
Hi,

can anyone please explain to me why convolution in $\displaystyle L^1 $ does not have an identity element?

i.e.

there is no $\displaystyle u \in L^1(R^n)$ such that

$\displaystyle u * g = g$ $\displaystyle \forall g \in L^1(R^n) $

Thanks - Aug 22nd 2008, 11:13 AMOpalg
An informal, "physicist's", explanation would go like this. If g*u=g for all g then $\displaystyle g(x) = \int_{\mathbb R^n}g(x-t)u(t)\,dt$. But you would not expect the value of g at x to depend on the values of g at points other than x itself. Therefore the function u must be zero everywhere except at 0, where it has to be so infinitely huge that the value of the integral is g(x). In other words, u has to be the Dirac delta function. But this is not an element of $\displaystyle L^1(\mathbb R^n)$ (and in fact is not a function at all).

If you want a more mathematical argument, then you could try something like this. let $\displaystyle \varepsilon>0$ and define f by $\displaystyle f(x) = \begin{cases}1&\text{if }|x|<\varepsilon,\\0&\text{otherwise}.\end{cases}$ Then $\displaystyle f(x) = (f*u)(x) = \int_{\mathbb R^n}f(x-t)u(t)\,dt = \int_{|x-t|<\varepsilon}u(t)\,dt$. When |x|>ε that gives $\displaystyle \int_{|x-t|<\varepsilon}u(t)\,dt = 0$. You can probably believe (and you may even be able to prove) that this implies that u(t)=0 whenever |x|>ε. Since ε is arbitrary, it follows that u is zero everywhere except at 0.

However, $\displaystyle L^1(\mathbb R^n)$ does have what is called an*approximate identity*. That is a sequence of elements $\displaystyle u_k$ of norm 1 such that $\displaystyle f*u_k\to f$ (in the $\displaystyle L^1$-norm) as k→∞, for all $\displaystyle f\in L^1(\mathbb R^n)$. For example, when n=1 you can take $\displaystyle u_k(x) = {\textstyle\frac k{\sqrt{\pi}}}e^{-k^2x^2}$. This gives a sequence of functions that have increasingly large sharp spikes at the origin and are very small elsewhere. - Aug 22nd 2008, 11:32 AMmcbain666
Thanks,

both explanations are pretty convincing.

I've been trying with characteristic functions but haven't managed to prove the claim.

By the way, if

$\displaystyle \frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r} $ where $\displaystyle p,q,r \in (1,\infty) $,

how could one prove that if

$\displaystyle f \in L^p, g \in L^q $

then

$\displaystyle f * g \in L^r $

?

I've been trying with Holder but can't seem to get anywhere, I really should need a good functional analysis book...

Thanks again for your help. - Aug 22nd 2008, 12:27 PMOpalg
I believe this is usually proved by using interpolation theorems, as discussed here.

By the way, if you want to ask a new question, you should really start a new thread. That makes it easier for other people to access it. - Aug 22nd 2008, 02:22 PMmcbain666
Thanks