Fourier series: coefficients, convergence and Parseval

On \( [-\pi, \pi] \) the functions \( 1, \cos x, \sin x, \cos 2x, \sin 2x, \dots \) are mutually orthogonal: \( \int_{-\pi}^{\pi}\sin mx\sin nx\,dx = \pi\delta_{mn} \) for \( m, n \ge 1 \), similarly for cosines, and every sine is orthogonal to every cosine. Assuming \( f(x) = \frac{a_0}{2} + \sum_{n\ge1}(a_n\cos nx + b_n\sin nx) \) and integrating against each basis function isolates one coefficient: \[ a_n = \frac1\pi\int_{-\pi}^{\pi} f(x)\cos nx\,dx, \qquad b_n = \frac1\pi\int_{-\pi}^{\pi} f(x)\sin nx\,dx. \] For \( f(x) = x \) (an odd function, so \( a_n = 0 \)) integration by parts gives \( b_n = 2(-1)^{n+1}/n \), which the example computes.

Convergence needs care, and there are three kinds. For every square-integrable \( f \) the partial sums converge in mean square: \( \int|f - S_N|^2 \to 0 \). If \( f \) is piecewise continuously differentiable, the series converges at every point, to \( f(x) \) where \( f \) is continuous and to \( \tfrac12(f(x^-) + f(x^+)) \) at a jump. If the \( 2\pi \)-periodic extension of \( f \) is continuous and piecewise \( C^1 \), the convergence is uniform. For \( f(x) = x \) the periodic extension jumps at \( \pm\pi \), the series gives \( 0 \) there, and near the jump the partial sums overshoot by about 9% of the jump however many terms you take: the Gibbs phenomenon.

Parseval's identity is Pythagoras in infinitely many dimensions: \[ \frac1\pi\int_{-\pi}^{\pi}|f|^2\,dx = \frac{a_0^2}{2} + \sum_{n\ge1}\big(a_n^2 + b_n^2\big). \] For \( f(x) = x \) the left side is \( 2\pi^2/3 \) and the right is \( \sum 4/n^2 \), which proves \( \sum 1/n^2 = \pi^2/6 \).

Picture it: add sine waves one at a time and watch a sawtooth assemble, each new term fixing finer detail, with a persistent ear of overshoot at the jump that narrows but never shrinks in height.

Think it: smoothness is decay. If the periodic extension of \( f \) has \( k \) continuous derivatives then integrating by parts \( k \) times shows the coefficients are \( o(n^{-k}) \); conversely, fast decay of coefficients means smoothness. Measuring a function by \( \sum (1 + n^2)^s(a_n^2 + b_n^2) \) is how Sobolev spaces are defined on a circle, and it is why the heat equation, which multiplies coefficients by \( e^{-n^2t} \), makes everything smooth.