mean-square convergence
/ Parseval: par-suh-VAL /
Pointwise convergence checks the series one point at a time, which can be fussy near jumps. Mean-square convergence asks an easier, more forgiving question: not 'is the series right at every single point?' but 'is the total squared error, averaged over the whole interval, shrinking to zero?' It measures the partial sum's miss by the area of its squared difference from f — and that area can go to zero even while a few isolated points (like the tops of Gibbs spikes) never quite settle.
Precisely, the partial sum S_N converges to f in mean square if the integral of (f - S_N)^2 over the interval tends to zero as N grows. The beautiful fact is that this holds for EVERY square-integrable function f — far more functions than pointwise convergence reaches, with no smoothness required at all. It rests on completeness: the sines and cosines are not just orthogonal, they are a complete orthogonal basis, so nothing in the L^2 space is left unrepresented. There is no nonzero function orthogonal to all of them.
Completeness is captured by Parseval's identity: the integral of f^2 over the interval equals the sum of the squares of the Fourier coefficients (suitably normalized). Read physically, this is conservation of energy — the total energy of the signal equals the sum of the energies carried by its individual frequency components. So a Fourier series doesn't lose or create energy; it just redistributes f's energy across modes. Mean-square convergence plus Parseval is the cleanest, most robust sense in which a Fourier series equals its function, and it is the natural setting for the modern theory of PDEs.
For the square wave with series (4/pi)[ sin x + (1/3) sin 3x + (1/5) sin 5x + ... ], Parseval's identity says the sum of the squared coefficients, (4/pi)^2 [ 1 + 1/9 + 1/25 + ... ], equals the mean square of the function. Working it out yields the classic value 1 + 1/9 + 1/25 + ... = pi^2/8 — energy bookkeeping handing you a number-theory sum for free.
Parseval's identity equates the energy of f with the total energy spread across its Fourier modes.
Mean-square convergence does NOT imply convergence at every point — a partial sum can be tiny in total squared error yet still spike near a jump (Gibbs). The three notions of convergence (pointwise, uniform, mean-square) are genuinely different and neither implies the others in general.