f, continuous, piecewise differentiable with f ' bounded ... fourier series
- From: G Patel <gaya.patel@xxxxxxxxx>
- Date: Wed, 1 Apr 2009 07:30:15 -0700 (PDT)
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)?
Thank you
.
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