Manuscript En01-B@2 · En01-B-v2.tex · section · f226d1eb

Gaussian regularization: sign, margin, and validated integration

Gaussian regularization: sign, margin, and validated integration · lines 1644–1662

En01-B@2#eq:b

Set

α = r_{c} / R, v = (1/4, 1/2, 3/4), b (α) = B_{r_{c}} / K_{G} .

Equation (eq:spectral) gives

b (α) = \frac{2}{π} \int_{0}^{\infty} e^{- α^{2} u^{2}} F (u)^{2} S (u) d u, S (u) = - i \sum w_{i} sinc (v_{i} u) .

The constant term vanishes, and $S (u) = O (u^{4})$ at the origin. Its pointwise oscillations do not determine the sign of the complete integral.

Displayed formulas in this passage

En01-B@2#eq:b · line 1653
$b (α) = \frac{2}{π} \int_{0}^{\infty} e^{- α^{2} u^{2}} F (u)^{2} S (u) d u, S (u) = - i \sum w_{i} sinc (v_{i} u) .$
unlabeled · line 1647
$α = r_{c} / R, v = (1/4, 1/2, 3/4), b (α) = B_{r_{c}} / K_{G} .$