convergence in the mean

Pointwise convergence asks the partial sums to be right at every single point — a demanding, point-by-point standard. Convergence in the mean asks something gentler and more robust: that the total squared error between f and its partial sum, integrated over a period, shrinks to zero. It measures average closeness, not closeness everywhere.

Formally, S_N -> f in the mean if the L2 norm of (f - S_N) tends to 0, that is integral over [-pi, pi] of |f(x) - S_N(x)|^2 dx -> 0 as N -> infinity. This is exactly convergence in the Hilbert space L2 of square-integrable functions, where the Fourier partial sums are the orthogonal projections onto the span of the first frequencies — and orthogonal projection is always the best L2 approximation.

Here the theory is beautifully complete, with none of the pathology of the pointwise case. For every f in L2, its Fourier series converges to f in the mean — full stop, no extra hypotheses. The trigonometric system is a complete orthonormal basis of L2, so Parseval's identity holds and the partial sums are optimal. The price for this clean statement is that mean convergence says nothing about any individual point: two functions agreeing in the mean may differ on a set of measure zero.