Partial sum = convolution with the Dirichlet kernel
Substitute the coefficient integrals into S_N f(x) and swap sum and integral. The tones collapse into a single closed-form weight, the Dirichlet kernel D_N. The partial sum is then a convolution: S_N f(x) = (f * D_N)(x). Convergence of Fourier series is really a question about how the kernels D_N behave.
D_N(t) = sum_{n=-N}^{N} e^{i n t}. This is a finite geometric series with ratio e^{i t}:
D_N(t) = e^{-iNt} * ( e^{i(2N+1)t} - 1 ) / ( e^{it} - 1 ).
Factor e^{i t/2} top and bottom to symmetrize:
D_N(t) = ( e^{i(N+1/2)t} - e^{-i(N+1/2)t} ) / ( e^{it/2} - e^{-it/2} )
= sin((N + 1/2) t) / sin(t/2).
Key facts:
(1) (1/2pi) integral over [-pi,pi] D_N(t) dt = 1 (the n=0 term integrates to 2pi, the rest to 0).
(2) D_N CHANGES SIGN and oscillates; it is NOT >= 0.
(3) Its L^1 size grows: (1/2pi) integral |D_N(t)| dt ~ (4/pi^2) log N -> infinity.
These 'Lebesgue constants' diverging is the seed of every pointwise pathology to come.Why pointwise convergence is genuinely hard
Because ‖D_N‖₁ → ∞, the maps f ↦ S_N f(0) are unbounded as N grows. By the uniform boundedness principle there exists a continuous f whose Fourier series diverges at a point. So pointwise convergence of Fourier series can fail even for continuous functions — continuity alone is not enough. Positive results need extra smoothness; the clean sufficient condition is Dirichlet's, below.
Fejér: average the partial sums and everything works
The cure is to stop summing and start averaging. The Cesàro mean σ_N f = (S_0 f + … + S_{N−1} f)/N is convolution with the Fejér kernel K_N. Unlike D_N, the Fejér kernel is non-negative — and that single sign change turns it into an approximate identity, a peak that concentrates at 0 while keeping mean 1.
Fejer kernel: K_N(t) = (1/N) sum_{m=0}^{N-1} D_m(t) = (1/N) ( sin(Nt/2) / sin(t/2) )^2 >= 0.
Three properties making {K_N} an approximate identity:
(i) K_N(t) >= 0 (a square -- positivity!)
(ii) (1/2pi) integral over [-pi,pi] K_N = 1 (mean one)
(iii) for fixed delta > 0, (1/2pi) integral_{delta <= |t| <= pi} K_N(t) dt -> 0 (mass piles up at 0)
Fejer's theorem: if f is continuous and 2pi-periodic, then sigma_N f -> f UNIFORMLY.
Proof sketch: sigma_N f(x) - f(x) = (1/2pi) integral K_N(t) [ f(x-t) - f(x) ] dt.
Split into |t| < delta and |t| >= delta.
Near piece: |f(x-t) - f(x)| < epsilon by UNIFORM CONTINUITY of f, and integral K_N <= 1, so < epsilon.
Far piece: |f(x-t)-f(x)| <= 2 max|f|, times (iii) which -> 0; choose N large.
Both bounds are independent of x => UNIFORM convergence. QED