Homework for holidays - 1

[Camillo]

Define for which $alpha in RR $ may exist solution in $L^1(RR) $ of the equation:

$y+alpha y$*$e^(-x^2) =f $, being $f in L^1(RR) $ assigned ,$ dot f inL^1(RR)$ [$dot f $ means Fourier transform of $f $ and * means convolution].
Express $y $ in integral form

[Kroldar]

"elgiovo":Are you sure? I'm probably wrong, but I tried calculating the Laurent series for $g(xi)=1/(1-sqrt(pi) e^(-(xi^2)/4))$ (obtained with $alpha=-1$) about $sqrt(2 *ln(pi))$, obtaining a pole of order 1.

That was my mistake, sorry. I could not generalize a result I found, then.

[elgiovo]

"Kroldar":the function $1/(1+sqrt(pi)alpha e^(-(xi^2)/4))$ would have a pole of order 2 and could be considered (in such a way) a tempered distribution

Are you sure? I'm probably wrong, but I tried calculating the Laurent series for $g(xi)=1/(1-sqrt(pi) e^(-(xi^2)/4))$ (obtained with $alpha=-1$) about $sqrt(2 *ln(pi))$, obtaining a pole of order 1.
What are the conditions for $g(xi) in ccS'(RR)$? I've found that $g(xi)$ is a tempered distribution iff $|langle g, phi rangle |<=C*p(phi)=C*"sup"_(xi in RR)[xi*(del g(xi))/(del xi)]$, $phi in ccS$ (in $N$ dimensions, the functions $p_(alpha,beta)(phi)="sup"_(xi in RR^N)[|x^(alpha)D^(beta)phi(xi)|]$ define a family of seminorms in $ccS(RR^N)$, for every multi-index $alpha,betain NN_0^N$).

[Kroldar]

"elgiovo":
Otherwise we cannot conclude only from $dot y (xi) in C(RR)$. In this occasion we obtain the same result, but it may be a fortuitous event.

Of course, in fact the exercise asked for which $a$ may exist solutions in $L^1$... I read the question this way: which are the necessary conditions for existence of $y(x) in L^1$?
Moreover, if $dot y(xi) !in L^1$ it could be antitransformed by considering the principal value integral.

"elgiovo":I think this is interesting. I'll investigate a little bit.

If you realize anything interesting, tell us!

[elgiovo]

"Kroldar":in my opinion, we just could assume $dot y(xi)$ continuous (the other properties are obvious).

We have to take the inverse Fourier transform (which is almost exactly the same thing as the Fourier transform) of $dot y(xi)$. We have the implication $dot y(xi)in L^1(RR) => y(x)in C(RR)$ and $lim_(x to +- infty)y(x)=0$. But does the converse hold? I.e.: is $=>$ actually $<=>$? Otherwise we cannot conclude only from $dot y (xi) in C(RR)$. In this occasion we obtain the same result, but it may be a fortuitous event.

"Kroldar":What do you think about?

I think this is interesting. I'll investigate a little bit.

[Kroldar]

I only think that the exercise was not so difficult and could be enough to verify the well-know properties of the Fourier transform of a summable function. So, in my opinion, we just could assume $dot y(xi)$ continuous (the other properties are obvious).

I guessed you had a larger vision of this exercise, and that's good. I also agree with your arguments, a posteriori.

Then you asked what happens if we consider the space of tempered distributions... Very good question! I'm not sure, but I think that may exist solutions also for $a <= -1/sqrt(pi)$, in fact in that case the function $1/(1+sqrt(pi)alpha e^(-(xi^2)/4))$ would have a pole of order 2 and could be considered (in such a way) a tempered distribution (let's remember that the Fourier transform of a tempered distribution is still e tempered distribution). And you? What do you think about?

[elgiovo]

These are my ideas:

- We're looking for $alpha$ s.t. may exist a solution $y(cdot)inL^1(RR)$.

- If we let $dot y(xi) in L^2(RR)$ then the result will be $y(x) in L^2(RR)$, because of the Parseval theorem. Being in $RR$, we can't say whether or not $y(cdot)inL^1(RR)$ a priori.

- $dot y(xi)$ is the product of two functions, which implies $y(x)=(f@g)(x)$, being $g=ccF^(-1)[(1+sqrtpi alpha e^(-(xi^2)/4))^(-1)]$, and we know that $dot f(cdot)$, $f(cdot) in L^1(RR)$. Two theorems came across my mind:

1) $finL^1(RR)$, $gin L^1(RR)$ $rarr$ $y=f@g in L^1(RR)$.
2) $dot f in L^1(RR)$, $dot g in (L^1(RR))'=L^(oo)(RR)$ $rarr$ $dotf dot g in L^1(RR)$.

So I've thought that we can hope in the existence of a $L^1(RR)$ solution if $dot f dot g in L^1(RR)$, especially because $dot g !in L^p(RR)$, $forall$ $1<=p But unluckily neither the domain nor the range of the Fourier transform are well defined. For example, what if we considered the set of Schwartz functions or the space of tempered distributions? This subject is of interest to me.

[Kroldar]

"elgiovo":
For the existence of a $L^1(RR)$ solution we need $dot y(xi)in L^1(RR)$

Why? If $dot y(xi) !in L^1$ it could be $y(x) in L^1$.
Instead $y(x) in L^1 => dot y(xi)$ continuous (and bounded) and $0$ for $xi to -+oo$.

[elgiovo]

Taking the Fourier transform of the equation, we get

$doty(xi)+sqrt(pi)alpha dot y(xi)e^(-(xi^2)/4) =dot f(xi)$,

i.e.

$dot y(xi)= dot f (xi)* 1/(1+sqrt(pi)alpha e^(-(xi^2)/4))$.

For the existence of a $L^1(RR)$ solution we need $dot y(xi)in L^1(RR)$; by Hölder inequality, we have

$dotf(xi)inL^1(RR)$ $^^$ $(1+sqrt(pi)alpha e^(-(xi^2)/4))^(-1)in (L^1)'(RR)=L^(oo)(RR) rarr dotf(xi) *(1+sqrt(pi)alpha e^(-(xi^2)/4))^(-1) in L^1(RR)$.

The second condition may be written as

$alpha sqrt(pi)e^(-(xi^2)/4) ne -1$ $forall xi$ $rarr$ $-1/(alpha sqrt pi) > 1$ $vv$ $-1/(alpha sqrt pi )<0$ $rarr$ $alpha> -1/(sqrt pi)$.

Now we're able to write the solution in integral form:

$y(x)=1/(2pi) int_(-oo)^(oo) dotf(xi) * (1+sqrt(pi)alpha e^(-(xi^2)/4))^(-1) * e^(i xi x)"d"xi=(f(x))/(2pi) @ int_(-oo)^(oo)(1+sqrt(pi)alpha e^(-(xi^2)/4))^(-1) * e^(i xi x)"d"xi=1/(2pi)int_(-oo)^(oo) int_(-oo)^(oo) f(x- zeta)*(1+sqrt(pi)alpha e^(-(xi^2)/4))^(-1) * e^(i xi zeta) "d"xi"d"zeta$.