- #1
parton
- 83
- 1
I have a problem understanding the following:
I should calculate the Fourier transform of a product of three functions:
[tex] \mathcal{F} \left[ f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) \right] = \int dx_{1} dx_{2} f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) e^{i p x_{1} + i q x_{2}} [/tex]
okay, and this goes over into a convolution:
[tex] = \dfrac{1}{2 \pi} \hat{f}(p) \ast \hat{g}(q) \ast \hat{h}(p+q) = \dfrac{1}{2 \pi} \int dk \hat{f}(p-k) \hat{g}(q-k) \hat{h}(k) [/tex].
I know how to calculate the convolution between two functions, but here we have three and I don't undestand how to get to the last line here. Could somebody explain that to me, please?
I should calculate the Fourier transform of a product of three functions:
[tex] \mathcal{F} \left[ f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) \right] = \int dx_{1} dx_{2} f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) e^{i p x_{1} + i q x_{2}} [/tex]
okay, and this goes over into a convolution:
[tex] = \dfrac{1}{2 \pi} \hat{f}(p) \ast \hat{g}(q) \ast \hat{h}(p+q) = \dfrac{1}{2 \pi} \int dk \hat{f}(p-k) \hat{g}(q-k) \hat{h}(k) [/tex].
I know how to calculate the convolution between two functions, but here we have three and I don't undestand how to get to the last line here. Could somebody explain that to me, please?
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