### III.27 The Fourier Transform

*Terence Tao*

Let *f* be a function from to . 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, *f*_{e}, and an odd part, *f*_{0}. For instance, the function *f*(*x*) = *x*^{3} + 3*x*^{2} + 3*x* + 1 is neither even nor odd, but it can be written as *f*_{e}(*x*) + *f*_{0}(*x*), where *f*_{e}(*x*) = 3x^{2} + 1 and *f*_{0}(