Re: Fourier analysis problem on which I'm stumped

On Apr 9, 6:59 pm, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On 9 Apr 2007 12:05:59 -0700, "Snis Pilbor" <snispil...@xxxxxxxxx>
wrote:

The problem is this: "Suppose f is in L^1([-1/2,1/2]),
f(-1/2)=f(1/2). Let A subset [-1/2,1/2] be such that f is bounded on
A and for any epsilon>0, there is delta>0 such that for all x in A,
integral from -delta to delta of |(f(x+y)-f(x))/y| dy, is < epsilon.
Then prove that for any epsilon>0 there exists tau>0 such that

"tau" does not appear below. What's the actual statement of the
problem?



Argh, forgive me. Of course the blurb
"|x_1-x_2|<epsilon" ought to be "|x_1-x_2|<tau". Thank you for your
patience.



for all
x_1,x_2 in A, with |x_1-x_2|<epsilon, we have
integral from -1/2 to 1/2 of | (f(x_1+y)-f(x_1)-f(x_2+y)+f(x_2)) / y |
dy, is < epsilon."

Well, of course with the hypotheses we have, we can split the integral
into an integral over (-delta,delta) plus one over (-1/2,-delta) union
(delta,1/2), such that the integral over (-delta,delta) is as small as
we like. And then the point is, in the integral on (-1/2,-delta)
union (delta,1/2), the 1/y is bounded by a constant. So it comes down
to showing we can pick tau>0 such that
int_{-1/2}^{1/2} |f(x_1+y)-f(x_2+y)-f(x_1)+f(x_2)| dy
is arbitrarily small. But I'm having tons of trouble showing this...

Thanks for any help.

SP

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David C. Ullrich

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