9. Series of Functions & Fourier

When series of functions converge uniformly — plus Fourier series and the Gibbs phenomenon.

• The Weierstrass M-Test — One test that settles uniform convergence of a series of functions all at once. The key is a cap that dominates each term at every point. If the caps themselves sum finitely, the tail shrinks across the whole interval together. So the error band tightens everywhere at once. For example, think of 📦 box lids covering every item. If the lid sizes add up finitely, nothing can poke out anywhere.
• Dini's Theorem — When does pointwise convergence of a monotone family upgrade to uniform? On a closed interval, it does. Monotonicity means no function falls behind once it catches up. The waiting times at all points tie into one finite bound. So the whole family closes in together. For example, think of climbing a 🪜 staircase. If nobody steps back, reaching the last step means the whole group has arrived.
• The Ringing That Never Dies — Round sine waves are layered to build a sharp square wave. More terms press the flat parts onto the wave. But the ringing beside each jump does not shrink. No number of terms removes that bump. For example, zoom into a 🏢 building in a 📷 photo. Smaller pixels sharpen the shape. Yet the jagged marks along the edges stay to the end.
• Riemann–Lebesgue Lemma — What happens to Fourier coefficients as oscillation speeds up? A rapidly wiggling wave averages away the function's bumps. Within one wavelength the function looks nearly constant. So as the frequency grows, every coefficient dies to zero. For example, shake a 🧵 string very fast. The wiggle blurs into a faint band, the ups and downs cancelling to leave only the middle.
• Fejér's Theorem — What happens when the ringing Fourier partial sums are averaged? Averaging the partial sums cancels their bumps against each other. The averaging kernel is positive and piled sharply at the center. So the averages settle onto the function across the whole interval at once. For example, overlay several shaky 📷 photos. Each photo's wobble cancels in the stack, and the picture turns clear.
• Parseval's Identity — Is a function's energy captured exactly on the coefficient side? The function's total square equals the coefficients' total square, as in the formula below. Adding modes pushes the captured share of energy toward all of it. Nothing is lost and nothing is invented. For example, count the 💰 coins in a 🐷 piggy bank by denomination and add them up. Changing how you count never changes the total.
• The Gibbs Phenomenon — Why does the bump beside a jump refuse to die? Zooming into the bump collapses every case onto one curve. Its first peak is fixed near 1.179. More terms narrow the bump but never lower it. For example, zoom into the corner of a 📷 photo. The jagged part gets narrower as you zoom, but how far it sticks out stays the same.
• Function Spaces: Cc, C0, Cb — This scene is about function spaces. See how the three families of continuous functions $C_c$, $C_0$, and $C_b$ nest and split. $C_b$ is all bounded continuous functions. $C_0$ holds the ones that fade to $0$ far away, and $C_c$ the ones that are exactly $0$ outside a finite interval. Both inclusions $C_c \subset C_0 \subset C_b$ are strict. For example, a ⛺ tent stands on a finite patch of ground. Fog thins out with distance but lies everywhere. An endless plain never fades away at all.