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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.27 The Fourier Transform

Terence Tao

Let f be a function from Image to Image. Typically, there is not much that one can say about f, but certain functions have useful symmetry properties. For instance, f is called even if f (-x) = f(x) for every x, and it is called odd if f (-x) = - f(x) for every x. Furthermore, every function f can be written as a superposition of an even part, fe, and an odd part, f0. For instance, the function f(x) = x3 + 3x2 + 3x + 1 is neither even nor odd, but it can be written as fe(x) + f0(x), where fe(x) = 3x2 + 1 and f0(

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