Re: fourier series of f ' relative to f
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Thu, 02 Apr 2009 05:09:14 -0500
On Wed, 1 Apr 2009 07:33:19 -0700 (PDT), G Patel
<gaya.patel@xxxxxxxxx> wrote:
If f is 2pi periodic, continuous, piecewise differentiable and f ' is
bounded, the how can I show:
c_n(f ') = i*n*c_n(f ) for nonzero n
Where c_n denotes nth Fourier coefficient.
Immediately when looking at this, it seems obvious that I "just" have
to differentiable both sides of:
f(t) = SUM(n=-inf to inf) c_n(f ) e^(int)
which gives
f '(t) = SUM(n=-inf to inf) i*n*c_n(f ) e^(int)
And by uniqueness of fourier series representation, c_n(f ') =
i*n*c_n
(f )
I believe this would be a perfectly good proof if the SUM was a
finite
SUM, correct?
But in this case I have to justify differentiating inside a infinite
sum, correct?
What is the theorem that can allow me to justify moving
differentiation operator inside infinite sums (or infinite limits)?
There exist theorems that allow you to differentiate
infinite sums term by term, but the hypotheses are
not going to be satisfied here.
Forget the sum - look at the _definition_ of
a_n(f').
Thank you
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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