Stephan Paukner :: syslog

Entries from Wednesday, February 1. 2006

Wednesday, February 1. 2006

d-dimensional Fourier transform

Master's Thesis

I know what the factor e^{2\pi\mathrm{i}x\omega} for x,\omega\in\mathbb{R} in the definition of the Fourier transform

\displaystyle(\mathscr{F}f)(\omega) = \int_{\mathbb{R}} f(x)e^{-2\pi\mathrm{i}x\omega}\,dx

means, if that factor is seen as a function in x: If \omega is fixed, then the variation of x delivers the circle function, returning to the starting point for every x\omega\in\mathbb{N}.

But what if x,\omega\in\mathbb{R}^d? Here, x\cdot\omega has to be understood as the inner product \langle x,\omega\rangle and the integral turns into the d-dimensional Lebesgue-integral, i.e., if x=(x_1,\ldots, x_d), this means

\displaystyle\int_{\mathbb{R}^d}dx = \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}dx_1\cdots dx_d

Now, what has to be understood by e^{2\pi\mathrm{i}\langle x,\omega\rangle} in higher dimensions?


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Posted by Stephan Paukner in Master's Thesis at 09:39 | Comments (0) | Trackbacks (0)
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