Convergence of Fourier Series

Fourier Analysis · Axiom Academy

LESSON Convergence of Fourier Series Understanding when and how Fourier series converge to functions Dirichlet established sufficient conditions that guarantee pointwise convergence of a Fourier series. These conditions are remarkably general and cover most functions encountered in practice. Piecewise continuity: f(x) is continuous except at finitely many points Finite discontinuities: At each discontinuity, both one-sided limits exist and are finite Bounded variation: f(x) has finitely many local maxima and minima 2. Convergence at Discontinuities At points of discontinuity, the Fourier series converges to the average of the left and right limits. This is a key feature of Fourier convergence. 3. Uniform vs Pointwise Convergence Uniform convergence is stronger than pointwise convergence. It means the series converges at the same rate everywhere, with no "bad" points where convergence is slower. For each x, S_N(x) → f(x) as N → ∞. Convergence rate may vary with x. sup_x |S_N(x) - f(x)| → 0 as N → ∞. Same convergence rate everywhere. 4. Mean-Square (L²) Convergence L² convergence measures the total squared error integrated over the entire domain. It's guaranteed for any square-integrable function, even with discontinuities. The partial sum S_N converges to f in L² if the integral of the squared difference goes to zero: 5. Summary of Convergence Types Different types of convergence provide different guarantees. Here's how they relate to each other and when each applies.

This is the written version of the interactive lesson above. See the full Fourier Analysis course.