Given is as function $ C(x)=A(x)B(x) $ and I want to find the connection between its fourier transforms. So I want to find a function $$ \tilde{C}(k)=\tilde{C}(\tilde{A}(k),\tilde{B}(k)) $$ ($\tilde{}$ stands for Fourier transformation). So basically i want to find out how to write$\tilde{A(x)B(x)}$ in a nice way. I wanted to use the Convolution Theorem $$ \digamma (A*B)= \digamma(A)\digamma(B)$$ (*stand for the convolution between A and B and here $\digamma$ stand for the Fourier Transformation)but there you only have the convolution of A and B fourier transformed and on the other side you have the product but that of the fourier transformed. I hope you can give me a tip.
2026-04-08 21:05:14.1775682314
Correlation between FT of function constisting of a product and FT of its factors
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I just had an idea. I can the $\digamma(A)$and$\digamma(B)$ as A and B and then the Theorem is $$ AB=\digamma(\digamma(A)*\digamma(B))$$ and if you FT then again you get $$\digamma(AB)=\digamma\digamma((\digamma(A)*\digamma(B)))=(\digamma(A)*\digamma(B))(-\bullet)$$ could i do that ?