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Unformatted text preview: FOURIER TRANSFORM TERENCE TAO Very broadly speaking, the Fourier transform is a systematic way to decompose generic functions into a superposition of symmetric functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin( nx ) or cos( nx )), and are often associated with physical concepts such as frequency or energy. What symmetric means here will be left vague, but it will usually be associated with some sort of group G , which is usually (though not always) abelian. Indeed, the Fourier transform is a fundamental tool in the study of groups (and more precisely in the representation theory of groups, which roughly speaking describes how a group can define a notion of symmetry). The Fourier transform is also related to topics in linear algebra, such as the representation of a vector as linear combinations of an orthonormal basis, or as linear combinations of eigenvectors of a matrix (or a linear operator). To give a very simple prototype of the Fourier transform, consider a realvalued function f : R R . Recall that such a function f ( x ) is even if f ( x ) = f ( x ) for all x R , and is odd if f ( x ) = f ( x ) for all x R . A typical function f , such as f ( x ) = x 3 + 3 x 2 + 3 x + 1, will be neither even nor odd. However, one can always write f as the superposition f = f e + f o of an even function f e and an odd function f o by the formulae f e ( x ) := f ( x ) + f ( x ) 2 ; f o ( x ) := f ( x ) f ( x ) 2 . For instance, if f ( x ) = x 3 +3 x 2 +3 x +1, then f e ( x ) = 3 x 2 +1 and f o ( x ) = x 3 +3 x . Note also that this decomposition is unique; there are no other even functions f e and odd functions f o such that f = f e + f o . This rudimentary Fourier transform is associated with the twoelement multiplicative group { 1 , +1 } , with the identity element +1 associated to the identity map x 7 x on the real line, and the other element 1 associated to the reflection map x 7  x . For a more complicated example, let n 1 be an integer and consider a complex valued function f : C C . If 0 j n 1 is an integer, let us say that such a function f ( z ) is a harmonic of order j if we have f ( e 2 i/n z ) = e 2 ij/n f ( z ) for all z C ; note that the notions of even function and odd function correspond to the cases j = 0 , n = 2 and j = 1 , n = 2 respectively. As another example, the functions z j , z j + n , z j +2 n , etc. are harmonics of order j . Then we can split any function f uniquely as a superposition f = n 1 j =0 f j of harmonics of order j , by means of the formula f j ( x ) := 1 n n 1 X k =0 f ( e 2 ik/n x ) e 2 ijk/n ; 1 2 TERENCE TAO note that the previous decomposition into even and odd functions was simply the n = 2 special case of this formula. The group associated to this Fourier transform is the n th roots of unity { e 2 ik/n : 0 k n 1 } , with each root of unity e 2 ik/n associated with the rotation...
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 Spring '08
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