Fourier transform

A Fourier series breaks a periodic function into discrete tones at integer frequencies. But what about a single non-repeating pulse — a function on the whole real line? As the period stretches to infinity, the spaced-out frequencies merge into a continuum, and the sum of coefficients becomes an integral. The Fourier transform is the result: it decomposes a function into a continuous spectrum of frequencies, telling you how much of each frequency it contains.

For an integrable function f on the real line, one common convention defines its transform as f-hat(xi) = integral over all reals of f(x) e^{-2*pi*i*x*xi} dx. The variable xi is frequency; f-hat(xi) is a complex number whose size is the amplitude and whose phase is the timing of that frequency. The transform converts hard analytic operations into easy algebraic ones: differentiation becomes multiplication by xi, and convolution becomes ordinary multiplication.

A foundational feature is the uncertainty principle: a function and its transform cannot both be sharply concentrated. A narrow spike in x spreads into a broad band of frequencies, and vice versa — squeezing one stretches the other. This is not a quirk of one example but a theorem (made precise by, e.g., the inequality of Heisenberg), and it underlies everything from quantum mechanics to signal processing. Note also that conventions differ: the placement of 2*pi and the sign in the exponent vary by author, so formulas must be read together with their normalization.