Dirichlet kernel
To study whether a Fourier series adds up to f, it helps to collapse the partial sum into a single integral. The Dirichlet kernel is the gadget that does this: it is the function you slide along and integrate f against to read off the N-th partial sum at a point.
Precisely, D_N(t) = sum over n from -N to N of e^{i n t} = sin((N + 1/2) t) / sin(t/2). The N-th partial sum of the Fourier series of f then equals the convolution (1/(2*pi)) integral over [-pi, pi] of f(x - t) D_N(t) dt. So convergence questions about partial sums become questions about how this kernel acts as N grows.
The decisive defect of D_N is that it is not non-negative: it oscillates and its overlap with sign-changes makes its L1 norm grow like (4/pi^2) log N, unboundedly. This L1 blow-up — the Lebesgue constants — is exactly why ordinary partial sums can misbehave, and why Fejér later replaced D_N with a positive averaging kernel to tame the convergence.
D_0(t) = 1, D_1(t) = 1 + 2 cos t, D_2(t) = 1 + 2 cos t + 2 cos 2t. Each integrates to 2*pi over [-pi, pi] (only the constant term survives), so a constant function is reproduced exactly — but the negative lobes of D_N grow as N increases.
The kernel integrates to 1 (after normalizing by 1/(2*pi)) but is not positive — the source of trouble.