Re: an entire function

In article <633c529f6kq6cmtkegp3vc1akpplcg74sg@xxxxxxx>,
David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:

On Mon, 01 May 2006 06:02:49 EDT, eugene <jane1806@xxxxxxx> wrote:

A have the following question:

Is there an entire function f(x)

You actually mean entire - analytic in the whole plane, right?

such that
int_R (f(x))^2dx < + infty and

You meant int_R |f(x)|^2dx < + infty, right?

the limit when x tends to + and - infinities is not 0.

If this means that the limit exists and does not equal
0 then the answer is obviously no.

If it means just "f(x) does not tend to 0 as x tends to
infinity" then the answer is actually yes, there does
exist such a function! (_Exactly_ how was the problem
stated?)

This surprises me - my first guess is that the answer
was no.

You may have forgotten the beautiful theorem of Carleman: Given a
continuous g : R -> R, and a continuous eps : R -> (0,oo), there
exists an entire f such that |f(t) - g(t)| < eps(t) for all t in
R.

When you hand this in make certain to say that
someone else did the problem for you. Also make certain
to fill in the details (what I write below should
convince people like WWW and RI that the answer is
yes, but your teacher will want more explanation):

Start with s(z) = sin(z)/z (and s(0) = 1). A function
of the form

f(z) = sum_j s(z+c_j)^{n_j}

satisfies the required conditions if n_j>0 and c_j>0
are chosen appropriately.

It's very easy to see that if n_j is large enough then
int_R |f(x)|^2 < infinity regardless of the choice of
c_j. And then it's easy to see that if c_j is chosen
properly f(x) does not tend to 0 at infinity. What
surprised me is that it's possible to choose c_j so
that f is entire.

Note first that sin(z) is bounded in every horizontal
strip. It follows that there exists a function
phiL R -> (0,infinity) such that phi(x) -> infinity
as x -> plus or minus infinity, such that s(z) -> 0
as z -> infinity withing the region Omega, defined
by Omega = {x+iy : |y| < phi(x)}. This shows that
s(z+c) -> 0 uniformly on compact subsets of the
plane as c -> 0...

Thanks


************************

David C. Ullrich
.

Relevant Pages
Loading