Computing $\sigma_N(f;t)$ from $s_n(f;t)$

[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!

 

[g_edgar]

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

 

[errordude]

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

?














?

 

[errordude]

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

 

[CRGreathouse]

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

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

 

[sjb-2812]

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

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?

 

[errordude]

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

lol

Spoiler

that was a good one.