Knowlpedia

Definition

Uniform parameter estimate

An estimate whose constants and validity domain are independent of the declared parameters.

The estimate F(ε,λ,x)=O(G(ε,λ,x))F(\varepsilon,\lambda,x)=O(G(\varepsilon,\lambda,x)), for G>0G>0, is uniform in (λ,x)∈Λ×K(\lambda,x)\in\Lambda\times K if there are a single CC and a single ε∗>0\varepsilon_*>0 such that

∣F(ε,λ,x)∣≤CG(ε,λ,x)|F(\varepsilon,\lambda,x)|\le C G(\varepsilon,\lambda,x)

for every 0<ε<ε∗0<\varepsilon<\varepsilon_* and every permitted (λ,x)(\lambda,x). This specifies the quantifiers hidden in .

Derivatives and a common domain

A smooth family estimate may assert, for every II, a constant CIC_I controlling ∂IF\partial^I F. The statement

∃ε∗  ∀I  ∃CI  ∀(ε,λ,x)\exists\varepsilon_*\;\forall I\;\exists C_I\;\forall(\varepsilon,\lambda,x)

on the same domain is stronger than allowing a different ε∗,I\varepsilon_{*,I} for each derivative order. Constants may depend on fixed profiles or a fixed construction stage while remaining uniform over scales and labels. These dependencies must be named.

References