Matpic
mathematics, in pictures
Series

Fourier series — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

    A square wave built entirely out of round ones

    Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

    part 1 · analysis
  2. The target, multiplied by one harmonic at a time. Four panels, each showing the square wave multiplied by a single sine. The areas cancel exactly except against the harmonics the wave actually contains.

    Where the coefficients come from

    The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

    part 2 · analysis
  3. The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

    When the period grows without bound

    A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

    part 3 · analysis
  4. A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

    The corners go first

    Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

    part 4 · analysis
  5. A plucked string, released. A string plucked into a tent shape at 0.5, drawn at times 0, 0.1, 0.25, 0.5, 0.75, 1: the corner splits into two corners that run apart, reflect off the ends upside down and meet again.

    A plucked string keeps its corners

    Heat smooths a sharp profile at once, because each harmonic decays at a rate set by the square of its frequency. Change one time derivative into two and nothing decays at all: each harmonic swings for ever, the corner of a pluck splits in two and runs along the string, and after one period the shape comes back exactly. The same sines, the same coefficients — and a flow that loses nothing.

    part 5 · analysis
  6. The Dirichlet kernel, a spike with ripples that do not die. The Dirichlet kernel for N = 4, 12 on the interval from −π to π: a central spike of height 2N + 1 and side ripples whose total area in absolute value grows with N.

    The ripples that make a series run away

    Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.

    part 6 · analysis

All series