Convergence of Fourier Series

Real Analysis · Axiom Academy

LESSON Convergence of Fourier Series When — and in what sense — does a Fourier series rebuild the function it came from, and what goes wrong at a jump? 1. The Series and Its Partial Sums For a -periodic function f , the Fourier series and its N -term partial sum S_N are: The full series — an infinite sum of harmonics The partial sum — the first N harmonics only Our running example is the square wave : +1 on and -1 on . It is odd, so only sine terms survive, and its real coefficients are for odd n (and 0 for even n ): 2. Pointwise Convergence — and the Midpoint Rule The weakest useful sense: fix a single point x and ask whether the numbers S_N(x) settle down as . f(x) wherever f is continuous , and 3. Uniform Convergence — One N for All x A much stronger demand: the same N must work everywhere at once, so the entire graph of S_N sits inside a thin band around f . When uniform convergence holds Each x gets its own N — the slow points can lag arbitrarily far behind. One N corrals the whole graph into the band at once. Strictly stronger. 4. Mean-Square Convergence — Always There is a third sense that costs the function almost nothing: measure the gap by the area of the squared error , not by its worst point. Right beside a jump the partial sum shoots past the function — and no matter how many terms you add, the overshoot stubbornly stays the same height. The overshoot holds at of the jump as — it never shrinks. The ear is pinned near , sliding toward the jump like 1/N .

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