# Sum derivation

1. Nov 17, 2009

### errordude

suppose, $$s_{n}(f;t) = \sum_{k=-n}^{n}\widehat{f}(k)e^{ikt}$$
and
$$\sigma_{N}(f;t)= \frac{1}{N+1}\sum_{n=0}^{N}s_{n}(f;t)$$.

how do i get from this $$\sigma_{N}(f;t)= \frac{1}{N+1}\sum_{n=0}^{N}s_{n}(f;t)$$.

to this

$$\sigma_{N}(f;t)= \sum_{n=-N}^{N}(1-\frac{|n|}{N+1})\widehat{f}(n)e^{int}$$

obviously one starts with:

$$\sigma_{N}(f;t)=\frac{1}{N+1}\sum_{n=0}^{N}\sum_{k=-n}^{n}\widehat{f}(k)e^{ikt}$$

thanks!

2. Nov 17, 2009

### g_edgar

And what happens when you reverse the order of summation ... the sum on k outside, the sum on n inside?

3. Nov 17, 2009

### errordude

?

?

4. Nov 17, 2009

### errordude

wow this must be slowest forum on the face of the planet

5. Nov 17, 2009

### CRGreathouse

Counterexample: http://www.myfinanceforum.com/

6. Nov 17, 2009

### sjb-2812

Perhaps, but remember we're not all free to check forums 25 hours a day, 8 days a week. Two hours 40 for what looks like a hint seems pretty good to me. Have you tried it?

7. Nov 18, 2009