A square wave is defined as f(x) = 1 for 0 < x < π and f(x) = −1 for −π < x < 0, with f(0) = 1. What does its Fourier series converge to at x = 0?
A1, because f(0) = 1 is the defined value of the function
B0, because the Fourier series converges to the average of the left and right limits: [f(0⁻) + f(0⁺)]/2 = [−1 + 1]/2 = 0
C−1, because the series tends toward the left-hand limit at a jump
DThe series diverges at jump discontinuities and has no value there
At a jump discontinuity, the Fourier convergence theorem states the series converges to the average of the left and right limits — regardless of the function's defined value at that point. Here f(0⁻) = −1 (approaching from the left) and f(0⁺) = 1 (approaching from the right), so the average is 0. The fact that f(0) was defined as 1 is irrelevant to convergence: the Fourier series 'splits the difference' at every jump, always producing the symmetric midpoint.
Question 2 Multiple Choice
As you add more and more terms (N → ∞) to the Fourier partial sums for a square wave, what happens to the overshoot near each jump discontinuity?
AThe overshoot disappears completely — with enough terms, the partial sums perfectly reproduce the square wave everywhere
BThe overshoot narrows to a shrinking region near each jump, but its height stays at approximately 9% of the jump magnitude and never vanishes
CThe overshoot grows larger as N increases, making the approximation worse near jumps
DThe overshoot disappears only if the function is redefined to equal the average at each jump point
This is the Gibbs phenomenon: the overshoot does not disappear — it merely concentrates. As N → ∞, the region of overshoot shrinks to an infinitesimally narrow band near each jump, but within that band the partial sums overshoot by approximately 9% of the jump height. The series does converge *pointwise* at the jump (to the average of the limits), but it does not converge *uniformly* near the jump — the maximum error over any small interval containing the jump remains bounded away from zero. Redefining f at the jump point changes nothing about the Fourier coefficients or the partial sums.
Question 3 True / False
For a piecewise smooth periodic function, the Fourier series converges at every point of continuity to the function's actual value there.
TTrue
FFalse
Answer: True
This is the central result of the Fourier convergence theorem under the Dirichlet conditions. If f is piecewise smooth (finitely many jumps and extrema per period) and continuous at a point x, then the partial sums Sₙ(x) → f(x) as N → ∞. The complication — convergence to the average rather than f(x) — only arises at jump discontinuities. At points of continuity, the Fourier series behaves exactly as one would hope.
Question 4 True / False
The Gibbs phenomenon shows that the Fourier series of a square wave fails to converge at jump discontinuities.
TTrue
FFalse
Answer: False
This is a common misinterpretation. The Fourier series of a square wave *does* converge pointwise at jump discontinuities — it converges to the average of the left and right limits. The Gibbs phenomenon is about the persistent overshoot near (but not at) the jump and the resulting failure of *uniform* convergence in a neighborhood of the discontinuity. These are distinct: pointwise convergence holds everywhere; uniform convergence fails near jumps. The overshoot narrows but never disappears, meaning the maximum error in any interval containing the jump stays bounded away from zero.
Question 5 Short Answer
Explain why the Gibbs phenomenon does not contradict the Fourier convergence theorem, and what it reveals about the nature of convergence at jump discontinuities.
Think about your answer, then reveal below.
Model answer: The Fourier convergence theorem guarantees pointwise convergence — for each fixed x, the partial sums Sₙ(x) converge to the correct limit (f(x) at continuity points, the average of limits at jumps). The Gibbs phenomenon operates at a different level: near a jump, the worst-case error across a small interval does not go to zero as N increases. This is a failure of *uniform* convergence, not pointwise convergence. The two notions are compatible: a sequence can converge pointwise everywhere while failing to converge uniformly on any interval containing a discontinuity. The Gibbs overshoot is the signature of this non-uniformity — it concentrates into an ever-narrower region but maintains constant height, an artifact of the global (sinusoidal) basis trying to approximate a local discontinuity.
Understanding the difference between pointwise and uniform convergence is essential for applications. In signal processing, the Gibbs phenomenon means that sharp transitions — square pulses, hard edges in images — cannot be perfectly reproduced by a finite Fourier representation. The ringing artifacts around sharp transitions in digital audio and image compression are direct consequences of this non-uniform convergence. The theorem assures correctness in the limit; the Gibbs phenomenon describes how that limit is approached.