Master's Thesis

Entries from February 2006

Monday, February 6. 2006

Understanding the Fourier transform

Master's Thesis

I borrowed the book “Fouriertransformation für Ingenieur- und Naturwissenschaften” by Klingen from the NuHAG library to use it as a help for understanding the (multidimensional) Fourier transform. Unfortunately, it seems to be on a rather applied/engineer level and only seems to contain one- and two-dimensional functions. We’ll see if it is useful to me.

Posted by Stephan Paukner in Master's Thesis at 19:59 | Comments (0) | Trackbacks (0)

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?

Continue reading "d-dimensional Fourier transform"

Posted by Stephan Paukner in Master's Thesis at 08:39 | Comments (0) | Trackbacks (0)

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