Fourier Transform - Fourier Transform On Euclidean Space

Fourier Transform On Euclidean Space

The Fourier transform can be in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions. For an integrable function ƒ(x), this article takes the definition:

where x and ξ are n-dimensional vectors, and x · ξ is the dot product of the vectors. The dot product is sometimes written as .

All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds. (Stein & Weiss 1971)