Manuscript Hawk02-E@1 · Hawk02-E.tex · corollary · 4841592b

Corollary: Explicit regulator selection

Reference-preserving finite approximation › Frequency discretization with exact packet identification · lines 1102–1130 · proof included or linked

Hawk02-E@1#cor:regulator-selection

Suppose $T > 0$ , $0 < b < 1$ , and $x = 2 GT$ . Choose an integer $m \geq 1$ such that

m + 1 \geq 4 e x^{2}, m \geq 2 + lo g_{2} (1/ b),

and choose the mesh so that

Δ \leq \frac{b}{T m} .

Then $b_{reg} (T; m, Δ) \leq b$ .

Proof. For $ℓ \geq m + 1$ , consecutive terms of (eq:Z-tail) have ratio at most $1/2$ . Also,

\frac{x ^{m + 1}}{( m + 1 )!} \leq (\frac{e x ^{2}}{m + 1})^{(m + 1) /2} \leq 2^{- (m + 1)} .

Thus $Z_{m} (x) \leq 2^{- m}$ , and $2 Z_{m} (x) \leq b /2$ . The mesh term is also at most $b /2$ .

Displayed formulas in this passage

unlabeled · line 1106
$m + 1 \geq 4 e x^{2}, m \geq 2 + lo g_{2} (1/ b),$
unlabeled · line 1112
$Δ \leq \frac{b}{T m} .$
unlabeled · line 1122
$\frac{x ^{m + 1}}{( m + 1 )!} \leq (\frac{e x ^{2}}{m + 1})^{(m + 1) /2} \leq 2^{- (m + 1)} .$