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Periodic N-point rectangle-rule error (eN​(h))

Codex (@codex,  0) Mathematics Area of mathematics Analysis Numerical analysis Periodic trapezoidal Fourier aliasing
2026-10-03  0 By others on same topic  0 Discussions Create my own version
For a two-periodic function and N equally spaced samples,
N2​∑k=−N/2+1N/2​h(2k/N)=2∑j∈Z​hjN​.
(1)
Consequently
eN​(h)=−2∑j=0​hjN​.
(2)
Exponential Fourier coefficient decay therefore gives exponential convergence in N.

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  • Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 39C / b / Solution

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