Fejér kernel

If the partial sums of a Fourier series jitter and overshoot, a natural fix is to average them — to take the running average of the first N partial sums rather than the raw N-th one. The kernel that produces this averaged sum is the Fejér kernel, and averaging works wonders precisely because this kernel, unlike Dirichlet's, is non-negative.

Concretely, F_N(t) = (1/(N+1)) sum from k = 0 to N of D_k(t) = (1/(N+1)) (sin((N+1) t/2) / sin(t/2))^2. Because it is a square, F_N(t) >= 0 everywhere; it integrates to 1 (after the 1/(2*pi) factor); and its mass concentrates near t = 0 as N grows. These three properties make F_N an approximate identity.

The payoff is Fejér's theorem: for any continuous 2*pi-periodic f, the Cesàro means (the convolutions of f with F_N) converge to f uniformly. Thus even when the plain Fourier series of a continuous function diverges somewhere, its averaged sums always recover the function. This is summability — a weaker, gentler kind of convergence that the troublesome Dirichlet kernel cannot deliver.