Fourier transform in n-dim Euclidean and Minkowski space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:47:38Z http://mathoverflow.net/feeds/question/93127 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93127/fourier-transform-in-n-dim-euclidean-and-minkowski-space Fourier transform in n-dim Euclidean and Minkowski space Wox 2012-04-04T14:35:39Z 2012-04-04T14:35:39Z <p>As far as I understood, the Fourier decomposition of a function $\boldsymbol{F}\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ where $\mathbb{R}^{n}$ is endowed with the Euclidean inner product $\left&lt;\cdot,\cdot\right>$ is given by</p> <p>$\boldsymbol{F}(\bar{x})=\int_{\mathbb{R}^{n}}{\tilde{\boldsymbol{F}}(\bar{\nu})e^{2\pi i \left&lt;\bar{\nu},\bar{x}\right>}}{d\bar{\nu}}$</p> <p>where $\tilde{\boldsymbol{F}}(\bar{\nu})=\int_{\mathbb{R}^{n}}{\boldsymbol{F}(\bar{x})e^{-2\pi i \left&lt;\bar{\nu},\bar{x}\right>}}{d\bar{x}}$</p> <p>How does this come about and for which functions does it apply? I'm not even able to find the right framework to work in (Hilbert spaces?). Secondly, could I just replace the Euclidean inner product by the Minkowskian inner product when in Minkowski space?</p>