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)
Read more about this topic: Fourier Transform
Famous quotes containing the words transform and/or space:
“The lullaby is the spell whereby the mother attempts to transform herself back from an ogre to a saint.”
—James Fenton (b. 1949)
“... the space left to freedom is very small. ... ends are inherent in human nature and the same for all.”
—Hannah Arendt (1906–1975)