Remainder estimate - Knowlpedia

For an approximation $S_\varepsilon$ to $F_\varepsilon$ , a remainder estimate bounds $R_\varepsilon=F_\varepsilon-S_\varepsilon$ , for example

\|R_\varepsilon\|_X\le C\varepsilon^a.

It must identify the norm or seminorm, the parameter range, exponent, and constant dependencies. In an expansion, $S_\varepsilon$ can be a finite truncation.

Higher order and flat errors

Relative to a nonzero term of order $\varepsilon^b$ , an error bound with $a>b$ is higher order. An error is flat in the parameter if, for every $N$ , it is $O(\varepsilon^N)$ in the specified topology. A single positive-order bound does not imply flatness, nor does a value bound imply estimates for derivatives.

Derivative topology

Infinite-order decay with derivatives makes the all-orders meaning precise for a positive scale function.