Fourier Transform 17 | Pointwise Convergence of Fourier Series

Fourier series don’t just converge in an average (L2) sense—they also converge point-by-point under a set of local “one-sided” smoothness conditions. For a 2π-periodic function f that is square-integrable (an L2 function), Fourier series are guaranteed to converge to f in the L2 norm, but that guarantee says nothing about what happens at individual points. Pointwise convergence requires extra structure, and the key insight here is that global continuity and differentiability are stronger than necessary: it’s enough that f behaves well from the left and from the right at the specific point where convergence is being tested.

Fix a point x0. The Fourier series at x0 converges if the left-hand and right-hand limits of f exist at x0, and if the corresponding one-sided derivatives also exist. Concretely, the left limit lim_{ε→0+} f(x0−ε) and the right limit lim_{ε→0+} f(x0+ε) must both exist; the function may jump, and the actual value f(x0) itself is irrelevant. The same one-sided requirement applies to the slope: the left derivative computed by approaching from x0−ε and the right derivative computed by approaching from x0+ε must exist as well. Under these conditions, the Fourier series converges pointwise at x0 to a specific number: if f is continuous at x0, the limit equals f(x0); if f has a jump at x0, the Fourier series converges to the midpoint (arithmetic mean) of the left and right limits. This is the classic “Gibbs-style” phenomenon in a precise convergence form: Fourier series can’t detect a single assigned value at a discontinuity, only the limiting behavior from either side.

To make the rule concrete, the discussion builds an explicit 2π-periodic function with a jump of size π at the origin. On one side it is constant, and on the other it is linear: on (0,π) it takes the form π−x, then the function is extended periodically. The jump at x=0 is between the left and right limiting values, and the midpoint prediction says the Fourier series should converge at x=0 to π/2. The transcript then computes the Fourier coefficients. The constant term comes from integrating the function over one period, yielding a0 = π/4. For k≠0, integration by parts produces coefficients of the form c_k involving terms like (−1)^k and 1/k^2, plus an additional factor proportional to 1/k. Using Euler’s formula, the complex Fourier series is rewritten into the standard real form with cosine and sine coefficients (a_k and b_k), where b_k simplifies to 1/k and a_k involves (1−(−1)^k)/k^2.

Finally, the Fourier series is assembled as a finite sum for each n and visualized numerically. The plot and the stated limit behavior confirm the theoretical prediction: as n grows, the partial sums approach π/2 at the origin, while the sine terms vanish at x=0 (since sin(0)=0) and the cosine terms remain. The payoff is twofold: a rigorous pointwise convergence criterion based only on one-sided limits and derivatives, and a practical way to extract explicit infinite-sum identities from Fourier series by evaluating them at special points like x=0.