Fourier analysis

For a sufficiently well-behaved function on the real line, define its transform by integrating against a complex oscillation. We use angular frequency k, a minus sign in the forward exponential, and a factor 1/(2π) in the inverse. Other conventions distribute factors differently or use cycles per unit length. None is inherently more correct, but mixing them produces incorrect amplitudes. Start with Schwartz functions, whose smoothness and rapid decay make the manipulations below safe.

A periodic function uses a discrete Fourier series because allowable modes must fit the period. A function on the full line uses a continuous transform. A discrete Fourier transform is another object: a finite invertible linear map on sampled data. Interpreting that data as samples of a continuous signal requires sampling assumptions. Aliasing can identify distinct continuous frequencies on the same sample grid, so a high-resolution plot of finite data does not recover frequencies that were never distinguished.

Integration by parts gives the transform of a derivative: multiplication by ik, provided the boundary term vanishes. Two derivatives give −k². This makes constant-coefficient differential equations especially simple. The heat equation becomes a first-order ordinary differential equation for each k, and each mode decays at a rate proportional to k². Short wavelengths disappear quickly because they have large frequency, explaining the smoothing effect without following every point of the original profile.

Convolution combines shifted copies of one function weighted by another. Its transform is the product of the transforms under our convention. Consequently a translation-invariant linear filter can be described by a multiplier in frequency space. Division by a multiplier requires caution where it vanishes or becomes small. An inverse filter may amplify tiny measurement errors, and a Green function may need a distributional interpretation or boundary prescription. Algebraic simplification can reveal ill conditioning rather than cure it.

Plancherel's theorem extends the transform from a convenient dense class to L2 while preserving its norm up to the chosen normalization. This means square-integrable signals can be handled even when the forward integral is not an ordinary absolutely convergent integral. Their transforms, like the original L2 elements, are defined up to changes on null sets. Equality of integrated energy does not automatically prescribe a value at every point or guarantee pointwise convergence of arbitrary partial reconstructions.

For periodic piecewise smooth data, Fourier partial sums approach the midpoint of a jump and exhibit Gibbs overshoot near it. Increasing the number of modes narrows the troublesome region without eliminating the limiting relative overshoot. Smoother summation methods, such as Fejér averaging, change convergence behavior. This is useful when interpreting field truncations and spectral simulations: the approximation rule and the function class both matter. A finite spectral cutoff is a controlled representation choice, not an exact description of every scale.