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I want to do the following calculation:

[tex]

c_{m,n} = \int_{-\infty}^{\infty} t^m \phi(t-n)\,dt

[/tex]

I know three things:

- I know the values of [tex]c_{m,n}[/tex] for m={0,1} and the first 4 for m={2,3}
- I know that my [tex]\phi[/tex] is symmetric, i.e. [tex]\phi(t) = \phi(-t)[/tex]
- The fourier transform of [tex]\phi(t)[/tex], i.e. [tex]\Phi(\omega)[/tex]

This is how I tried solving the problem:

[tex]

c_{m,n} = \int t^m \phi(t-n)\,dt =

\int t^m \phi(-(n-t))\,dt[/tex]

because of the symmetry

[tex] = \int t^m \phi(n-t)\,dt[/tex]

...is exactly the definition of a convolution...

[tex](t^m * \phi)(n) = \mathcal{F}^{-1}\left\{ \mathcal{F}\left\{t^m\right\} \Phi(\omega)\right\} = \mathcal{F}^{-1}\left\{ j^m \sqrt{2\pi} \delta^{(n)}(\omega) \Phi(\omega) \right\} =

j^m \sqrt{2\pi} \mathcal{F}^{-1}\left\{\delta^{(n)}(\omega) \Phi(\omega) \right\}

[/tex]

Now, writing the inverse fourier transform gives:

[tex]

j^m \sqrt{2\pi} \frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} \delta^{(n)}(\omega) \Phi(\omega) e^{j\omega n}\,d\omega = j^m \int_{-\infty}^{\infty} \delta^{(n)}(\omega) \underbrace{\Phi(\omega) e^{j\omega n}}_{f(\omega)}\,d\omega

[/tex]

Now it is well known that

[tex]\int \delta^{(n)}(x) f(x)\,dx = (-1)^n f^{(n)}(0)[/tex]

so that

[tex]

j^m \int_{-\infty}^{\infty} \delta^{(n)}(\omega) f(\omega)\,d\omega = j^m (-1)^m \left. \frac{\partial^m}{\partial \omega^m} f(\omega) \right|_{\omega = 0}

[/tex]

So the whole procedure should reduce to calculate the derivative of [tex]f(\omega)[/tex] and set the result to zero. I fill in my known [tex]\Phi(\omega)[/tex]:

[tex]

f(\omega) = \mathrm{sinc}^2 (\omega/2) \sqrt{ 1 - \frac{2}{3} \sin^2(w/2)} \cdot e^{j\omega n}

[/tex]

Problem 1: Even the 0th derivative is only computable via a limit. Calculating the limit with [tex]\omega \rightarrow 0[/tex] gives the result:

[tex]\sqrt{\frac{2}{2+\cos(2)}}[/tex]

But this is not true. The result should be 1 for [tex]m=0[/tex].

The result for [tex]m=1[/tex] should be linear: {0,1,2,3,...}. But as you can verifiy the result with my approach is constant.

Also the result for m={2,3} should give a a quadratic and cubic function. But with my approach it stays constant; Even worse, the solution is not always real!

.... So there is anywhere a mistake. Can anybody help me where?

Thank you,

divB

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# Integrating derivate of Dirac: Where is my fault?

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