Periodic trapezoidal Fourier aliasing

For a two-periodic Fourier series sampled at $2N$ equally spaced points, discrete averaging retains exactly the modes divisible by $2N$:If $|{\hat{h}}_n|\le M{c}^{|n|}$ with $0<c<1$, thenwhich is exponentially small.

For a two-periodic function and $N$ equally spaced samples,ConsequentlyExponential Fourier coefficient decay therefore gives exponential convergence in $N$.