Manuscript Em04-A@2 · Em04-A-v2.tex · remark · f5b28abc

Remark: Midpoint alternative

Compressed oscillator position measurements › Composite trapezoidal approximation · lines 1233–1244

The earlier midpoint construction is also valid as a different rule:

Q_{d}^{mid} = ℓ \sum h_{ℓ} G_{d} ((a_{ℓ} + b_{ℓ}) /2), η_{d}^{mid} = \frac{1}{24} ℓ \sum h_{ℓ}^{3} I_{ℓ} sup ∥ G_{d}^{''} ∥_{\infty} .

Taylor expansion at each midpoint proves this bound. A midpoint certificate must not be attached to a trapezoidal implementation.

Displayed formulas in this passage

unlabeled · line 1235
$Q_{d}^{mid} = ℓ \sum h_{ℓ} G_{d} ((a_{ℓ} + b_{ℓ}) /2), η_{d}^{mid} = \frac{1}{24} ℓ \sum h_{ℓ}^{3} I_{ℓ} sup ∥ G_{d}^{''} ∥_{\infty} .$