most recent 30 from http://mathoverflow.net 2013-06-19T13:55:39Z http://mathoverflow.net/feeds/question/62809 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62809/integral-transform-wanted Yrogirg 2011-04-24T05:50:00Z 2011-04-24T05:50:00Z

<p>Hello! Imagine I have a function</p> <p>$f(r_1, r_2) = \int_{-\infty}^\infty g(|r_1 - r_3|) g(|r_1 - r_4|) g(|r_2 - r_3|) g(|r_2 - r_4|) g(|r_3 - r_4|) \; d r_3 d r_4$</p> <p>Is there a way I could get rid of the integration via some (integral) transform?</p> <p>so transformed $f$, that is $\hat f (k_1, k_2)$, was some algebraic combination of $\hat g (k_1, k_2)$?</p> <p>Actually I need a bit more complicated case of $r$ being 3D vectrors, but I think the solution for a scalar version should be just the same</p> <p>$f(\boldsymbol r_1, \boldsymbol r_2) = \int g(|\boldsymbol r_1 - \boldsymbol r_3|) g(|\boldsymbol r_1 - \boldsymbol r_4|) g(|\boldsymbol r_2 - \boldsymbol r_3|) g(|\boldsymbol r_2 - \boldsymbol r_4|) g(|\boldsymbol r_3 - \boldsymbol r_4|) \; d \boldsymbol r_3 d \boldsymbol r_4$</p>