Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Defect-to-continuum bound

Continuum defect certification and the next accuracy target › Uniform defect conversion · lines 1775–1793 · proof included or linked

Em03-H@8#thm:defect

If $sup_{u} max (∣ d_{A} (u) ∣, ∣ d_{B} (u) ∣) \leq ϵ$ , then

u sup max (∣ A - A ∣, ∣ B - B ∣) \leq ϵ e^{1 + π /2} .

In particular, $∣ q - B (1) ∣ \leq ϵ e^{1 + π /2}$ .

Proof. Subtract the Volterra equations. The maximum component error satisfies

e (u) \leq ϵ + \int_{0}^{u} (1 + ∣ κ (s) ∣) e (s) d s .

Iterate the integral inequality. The exact nonnegative pulse has area $π /2$ .

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