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

Lemma: A cellwise derivative-enclosure interface

Compressed oscillator position measurements › Composite trapezoidal approximation · lines 1246–1269 · proof included or linked

Em04-A@2#lem:Gbound

On a cell $I$ , suppose validated bounds satisfy

I sup ∣ ϕ_{n}^{(j)} ∣ \leq U_{n, j}, c_{j} = (n = 0 \sum^{d - 1} U_{n, j}^{2})^{1/2}, j = 0, 1, 2.

Then

I sup ∥ G_{d}^{''} ∥_{\infty} \leq 2 c_{0} c_{2} + 2 c_{1}^{2} .

Proof. Differentiate to obtain

G_{d}^{''} = v_{d}^{''} v_{d}^{†} + 2 v_{d}^{'} (v_{d}^{'})^{†} + v_{d} (v_{d}^{''})^{†} .

A rank-one matrix $u v^{†}$ has operator norm $∥ u ∥ ∥ v ∥$ . The triangle inequality proves the bound.

Displayed formulas in this passage

unlabeled · line 1249
$I sup ∣ ϕ_{n}^{(j)} ∣ \leq U_{n, j}, c_{j} = (n = 0 \sum^{d - 1} U_{n, j}^{2})^{1/2}, j = 0, 1, 2.$
unlabeled · line 1256
$I sup ∥ G_{d}^{''} ∥_{\infty} \leq 2 c_{0} c_{2} + 2 c_{1}^{2} .$
unlabeled · line 1263
$G_{d}^{''} = v_{d}^{''} v_{d}^{†} + 2 v_{d}^{'} (v_{d}^{'})^{†} + v_{d} (v_{d}^{''})^{†} .$