Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

A remainder cannot be differentiated without another estimate

Transposition and execution audit › A remainder cannot be differentiated without another estimate · lines 3299–3315

For fixed nonzero $A \in D^{'}$ ,

r_{δ} = δ sin (δ^{- 2}) A

has a vanishing $O (δ)$ finite-order bound, but

δ \partial_{δ} r_{δ} = (δ sin (δ^{- 2}) - 2 δ^{- 1} cos (δ^{- 2})) A .

Thus a future family-level remainder is not automatically a scale-integrability estimate.

Displayed formulas in this passage

unlabeled · line 3302
$r_{δ} = δ sin (δ^{- 2}) A$
unlabeled · line 3306
$δ \partial_{δ} r_{δ} = (δ sin (δ^{- 2}) - 2 δ^{- 1} cos (δ^{- 2})) A .$