= Uniform error bound for nearby decaying exponentials
{title2=$\sup_{x\geq0}|e^{-ax}-e^{-bx}|\leq |a-b|/[e\min(a,b)]$}
For positive rates $a,b$, the <mean value theorem> in the rate gives a bound by $|a-b|x e^{-\min(a,b)x}$. Its maximum is $|a-b|/[e\min(a,b)]$. Combining this estimate with coefficient errors bounds a finite spectral sum uniformly on a half-line, including slow scales much longer than one. Positivity of both rates is essential: unstable exponentials need a different time-window or relative-error analysis.
Back to article page
