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Recipe

Determine whether a Fourier series can converge uniformly on an interval

Use when: you have a Fourier series and a specific interval, and need to know whether convergence there can be uniform

Goal: a yes-or-no conclusion for that interval, backed by a proof by contradiction whenever the answer is no

Taught in Why the distinction matters for a Fourier series specifically

Quick recipe

  1. Check the target function for discontinuities inside the interval
  2. Assume, for contradiction, that convergence is uniform there
  3. Apply the uniform-limit-of-continuous-functions fact
  4. Compare that forced conclusion against the known behavior of f
  1. 1

    Scan f, the function the series targets, for any jump discontinuity that falls inside the interval in question.

    Example

    The odd square wave jumps from -1 to 1 at t=0; an interval such as I=(-1,1) contains that jump.

    • the interval contains no jump of f — continue to Step 4; this argument raises no obstruction there.
    • the interval contains at least one jump of f — continue to Step 2 and build a contradiction around it.

    Why: uniform convergence of a continuous partial sum can only be threatened at a point where the target itself is discontinuous, so a jump is the only thing worth looking for.

  2. 2

    Suppose S_N\to f uniformly on the interval containing the jump. Note that every partial sum S_N is a finite sum of sines and cosines, so it's continuous everywhere, including at the jump.

    Why: this hypothesis is exactly what the rest of the argument is built to disprove.

  3. 3

    A uniform limit of functions continuous at a point is itself continuous there, so the Step 2 assumption forces f to be continuous at the jump.

    Example

    For the square wave, this step forces f to be continuous at t=0.

    Why: this is the general fact proved earlier in this section, now applied to the specific point at hand.

  4. 4

    Check whether f actually is continuous at the point Step 3 named.

    Example

    The square wave jumps from -1 to 1 at t=0, contradicting the "continuous at 0" conclusion from Step 3, so uniform convergence on any interval containing 0 is impossible. On I=[\pi/4,3\pi/4], which touches no jump, the argument never gets past Step 1's first case.

    • f really does jump there — contradiction, so uniform convergence on this interval is impossible.
    • no jump was found in Step 1 — nothing to contradict here; only Section 8's theorem for piecewise-smooth functions proves uniform convergence positively.

    Why: the contradiction itself is the whole proof; without an actual jump at that point, ruling convergence out this way isn't available, and proving convergence positively takes a different, stronger result.

Common mistakes

  • Concluding uniform convergence is impossible just because the partial sums look like they're oscillating, without checking whether f itself has a jump there.
  • Using this argument to try to prove uniform convergence positively: it only ever rules convergence out, at a jump.
  • Treating "no jump found" as proof of uniform convergence, instead of only the absence of this particular obstruction.
  • Checking only the interval's endpoints for a jump, instead of every point inside it.