Why does the Fourier series of the square wave converge pointwise but not uniformly?
The series converges pointwise because at every continuous point it approaches the function value, and at jumps it approaches the midpoint. However, it does not converge uniformly because the Gibbs overshoot near the jumps maintains a nonzero relative peak height regardless of how many terms are added, preventing the maximum error from vanishing globally.
Conditions
- The function has jump discontinuities.
- Convergence is analyzed over the entire period including the jumps.
Reasoning, step by step
- Verify pointwise convergence at continuous points.
- Verify pointwise convergence at jump points (midpoint rule).
- Examine the behavior of the error near the jumps.
- Note that the Gibbs overshoot peak does not decrease to zero.
- Conclude that uniform convergence fails due to the persistent overshoot.
Example
The script states: 'At continuous points the sums approach the original value. At a jump they approach the midpoint... Nearby overshoot narrows in location but its relative peak does not vanish. This distinguishes pointwise convergence from uniform convergence across the jumps.'
Common misconceptions
- Confusing pointwise convergence with uniform convergence.
- Believing that narrowing the overshoot location eliminates the error.
Watch the explanation
BilibiliFourier series
0:19 – 0:30Watch this moment ↗
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