The starting point for FOURIER ANALYSIS [III.27] is the observation that a wide class of periodic functions *f*(*θ*) with period 2π can be decomposed as infinite linear combinations of the TRIGONOMETRIC FUNCTIONS [III.92] sin*nθ* and cos*nθ*, or, equivalently, as sums of the form *a _{n}e^{inθ}*.

A useful way to think of a periodic function *f* defined on the real line is as an equivalent function *F* defined on , the unit circle in the complex plane. A typical point on the circle has the form e^{iθ}, and we define *F*(e^{iθ}) to be *f*(*θ*). (Note that ...

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