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

Theorem: Trapezoidal operator-error bound

Compressed oscillator position measurements › Composite trapezoidal approximation · lines 1210–1231 · proof included or linked

Em04-A@2#thm:trap, Em04-A@2#eq:traperror

The matrix in (eq:trapezoid) is positive and

Q_{d}^{trap} (B; L) - E_{d} (B \cap [- L, L])_{\infty} \leq η_{d}^{trap} (B; L) .

Proof. Positivity follows from positive weights. Twice integrating by parts, componentwise, gives for one cell

\int_{a}^{b} G_{d} (x) d x - \frac{h}{2} (G_{d} (a) + G_{d} (b)) = - \frac{1}{2} \int_{a}^{b} (x - a) (b - x) G_{d}^{''} (x) d x .

Since $\frac{1}{2} \int_{a}^{b} (x - a) (b - x) d x = h^{3} /12$ , taking operator norms and summing proves the claim.

Displayed formulas in this passage

Em04-A@2#eq:traperror · line 1213
$Q_{d}^{trap} (B; L) - E_{d} (B \cap [- L, L])_{\infty} \leq η_{d}^{trap} (B; L) .$
unlabeled · line 1224
$\int_{a}^{b} G_{d} (x) d x - \frac{h}{2} (G_{d} (a) + G_{d} (b)) = - \frac{1}{2} \int_{a}^{b} (x - a) (b - x) G_{d}^{''} (x) d x .$