(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine the Fourier transform on the tempered distribution:

[tex]

\langle f, \varphi \rangle

[/tex]

Where [tex] f[/tex] can be given by they taylor series representation:

[tex]

f = i\sum_{n=0}^{\infty} \frac {x^{3n+2}}{(2n)!}

[/tex]

3. The attempt at a solution

Fourier transform on tempered distribution is:

[tex]

F\langle f, \varphi \rangle = \langle F f, \varphi \rangle = \langle f, F \varphi \rangle

= i\int \sum_{n=0}^{\infty} \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx

= i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx

[/tex]

I'm stuck as how to resolve the infinite sum involving the Fourier transform of the test function [tex]\varphi[/tex].

Perhaps:

[tex]

i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx = i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\delta}(x-x_0) dx

= i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} e^{2{\pi}ikx_0}dx

= ie^{2{\pi}ik{x_0}}\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} dx

= ie^{2{\pi}ik{x_0}}\sum_{n=0}^{\infty}\frac {x^{3n+3}}{(2n)! * (3n+3)!}

[/tex]

Any help would be fantastically appreciated,

Jerry109

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# Homework Help: Fourier transform on a distribution

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