Recipe

Compute a Cesàro (Fejér) mean of a Fourier partial sum

Use when: you need a Fourier approximation that converges uniformly through a jump, with no Gibbs ringing, and can accept averaging across several partial sums instead of relying on a single one

Goal: the tapered trigonometric polynomial \sigma_N(t), ringing-free near any jump, built entirely from the same coefficients an ordinary partial sum already uses

Taught in Cesàro (Fejér) averaging as a fix for non-uniform convergence

1. 1

   Using f's own real-form Fourier coefficients, write out the ordinary partial sums S_0(t), S_1(t), \dots, S_N(t) — each one just keeps harmonics up through its own order and drops the rest, the plain-partial-sum definition Chapter 3 already gives.

   Example

   For the square wave (period 2\pi, odd, so only odd harmonics survive), the needed coefficients are Chapter 3's own result: b_1=\tfrac4\pi,\ b_3=\tfrac{4}{3\pi},\ b_5=\tfrac4{5\pi},\ b_7=\tfrac{4}{7\pi},\ b_9=\tfrac4{9\pi}, reused here rather than rederived.

   • a_n, b_n aren't found yet — first compute them with the ordinary coefficient recipe (Chapter 3, Section 1); everything below only reweights what that recipe already produces.
   • a_n, b_n are already known — skip straight to Step 2's averaging.

   Why: the Cesàro mean is built entirely out of the same coefficients an ordinary partial sum already uses — nothing new needs deriving, only re-averaging what's already known.

2. 2

   Form \sigma_N(t) = \dfrac{S_0(t)+S_1(t)+\cdots+S_N(t)}{N+1} — literally add the N+1 partial sums and divide by how many there are, the defining formula this section states.

   Example

   Averaging S_0 through S_9 (ten partial sums, so N=9) for the square wave builds \sigma_9 — the exact average the first worked example below carries out in full.

   ⚡ Shortcut

   Skip building all N+1 partial sums individually. Multiply each harmonic's own original coefficient directly by the linear taper weight \big(1-\tfrac{n}{N+1}\big) instead — re-expanding the defining average in terms of the original coefficients produces exactly this weight, so tapering once is equivalent to averaging N+1 partial sums by hand.

   Why: this average — not any single S_N on its own — is what Fejér's theorem actually promises converges uniformly with no overshoot.

3. 3

   Check that harmonic n's weight really is 1-\tfrac{n}{N+1}: weight 1 at n=0, tapering in a straight line down to weight \tfrac{1}{N+1} at n=N, and 0 for every harmonic beyond N.

   Example

   With N=9: n=1\to0.9, n=3\to0.7, n=5\to0.5, n=7\to0.3, n=9\to0.1 — matching the worked example below's tapered coefficients exactly.

   Why: a sharp on/off weight — full strength for n\le N, zero beyond it, exactly what a plain partial sum uses — is exactly what produces the ringing; the linear taper is the entire mechanism removing it.

4. 4

   Decide whether the interval you actually care about includes a jump or not — that answer, not a blanket preference, is what should choose between \sigma_N and S_N.

   • the target interval includes a jump and uniform convergence matters there — use \sigma_N, which converges uniformly through it with no overshoot.
   • the target interval stays away from every jump and fine detail matters more than smoothness — a plain S_N typically converges faster there, so prefer it instead.

   Why: Cesàro averaging isn't a strict upgrade over a plain partial sum — it trades away detail everywhere in exchange for guaranteed smoothness at a jump, so which one to reach for depends on where on the domain the accuracy actually needs to hold.