Re: Converse to Riemann-Lebesgue lemma

On Sun, 05 Apr 2009 20:43:22 +0100, Timothy Murphy
<gayleard@xxxxxxxxxx> wrote:

José Carlos Santos wrote:

The lemma (for Fourier series) states that
if |f(x)| is Lebesgue integrabls and periodic
then its Fourier coefficients a_n -> 0.

Is there a simple example of a sequence a_n -> 0
which does not arise from an L^1 function in this way?

A slightly different approach is: if the natural map F from L_1([0,1])
into c_0 happened to be surjective then, since it would be bounded
bijective between Banach spaces, it would be an isomorphism (by the
open mapping theorem). Therefore, there would be some M > 0 such that,
for each _c_ in c_0, ||F^{-1}(c)|| <= M.||c||, which is equivalent to
the assertion that, when _f_ in L_1([0,1]), ||F(f)|| >= M^{-1}.||f||.
It is easy to see that no such M can exist, using the functions _f_ of
the type f(x) = sum_{-n <= k <= n}exp(i k x).

I have to confess that this was an exam question here a year or two ago,
with the above solution suggested.
I was convinced that there must be a straightforward counter-example,

There's probably an explicit counterexample somewhere in Zygmund.

Giving an explicit example can't be all that easy - for example you
can't write down a sequence and say that it's not the Fourier
coefficients of an L^1 function because it tends to 0 too slowly,
because of the following curious fact: If c_n >= 0 and c_n -> 0
then there exists f in L^1 with f^(n) >= c_n for all n.

but it seems my colleague has one up on me.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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