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 inspired scribbler always has the gift for gossip in our common usage ... he or she can always inspire the commonplace with an uncommon flavor, and transform trivialities by some original grace or sympathy or humor or affection.”
—Elizabeth Drew (18871965)
“Not so many years ago there there was no simpler or more intelligible notion than that of going on a journey. Travelmovement through spaceprovided the universal metaphor for change.... One of the subtle confusionsperhaps one of the secret terrorsof modern life is that we have lost this refuge. No longer do we move through space as we once did.”
—Daniel J. Boorstin (b. 1914)