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        riddles >> medium >> Fourier Transform
        
(Message started by: MonicaMath on Feb 25^th, 2010, 7:38am)

Title: Fourier Transform
Post by MonicaMath on Feb 25^th, 2010, 7:38am

I have a question

I know that we can use the fact that (cosx)'=-sinx to find the fourier transform
for sinx on [-pi/2, pi/2] by using FT properties, But why we can't use the same idea for
(sinx)'=cosx to find the FT of cosx on [-pi/2, pi/2]

???

any help please

Title: Re: Fourier Transform
Post by Michael Dagg on Feb 25^th, 2010, 2:41pm

I think perhaps when you speak of sin with a Fourier transform on
[-pi/2, pi/2], you mean the function s(x) which agrees with sin x on
that interval and is extended to the line so that it has period pi. And
likewise for cos . If you work with the functions with period pi , you
look for a Fourier transform (series) that looks like this:

\sum_{k=-\infty} ^ {+\infty} c_k e^{2ikx}

I don't know just how you are proceeding, so I'll not get technical but
just do the easy thing and try to make one simple remark that may help.
When two functions have the same derivative, they differ by a constant,
which may be nonzero.

Note that s(x) has average value 0 , and hence 0th Fourier coefficient
0 , whereas c(x) has a nonzero average value and a nonzero 0th Fourier
coefficient.

Title: Re: Fourier Transform
Post by MonicaMath on Feb 28^th, 2010, 8:31am

not clear.... does anyone have another idea ?!

Title: Re: Fourier Transform
Post by Michael Dagg on Feb 28^th, 2010, 8:39pm

If you can show me how you are proceeding then
I think I can make it clear. You could start off by
demonstrating your first sentence.

Title: Re: Fourier Transform
Post by MonicaMath on Mar 1^st, 2010, 8:59am

I mean:
Is there any condition that should be satisfied on these functions so we can apply the result !? and if so, what is the condition that sinx (or cosx ) in
[-p1/2, pi/2] doesn't satisfied in this case ??

thank you

Title: Re: Fourier Transform
Post by Obob on Mar 1^st, 2010, 9:11am

That didn't clarify things at all.

Are you extending the functions periodically to the whole real line?

And you should provide the argument for how to calculate the FT of sin x.