Manuscript III@2 · III.tex · section · 82dad3e5

Spatial Fourier regularization

The unchanged free-product obstruction › Spatial Fourier regularization · lines 772–800

Use the unitary transform

s (k) = (2 π)^{- 3/2} \int_{R^{3}} e^{- i k \cdot x} s (x) d^{3} x .

Choose smooth $a, b : [0, \infty) \to [0, 1]$ with

a = 0 on [0, 1], a = 1 on [2, \infty),

b = 1 on [0, 1], b = 0 on [2, \infty) .

For $0 < 2 μ < Λ$ , let

χ_{μ, Λ} (k) = a (∣ k ∣/ μ) b (∣ k ∣/Λ), ρ = (μ, Λ),

and

N_{ρ} s (k) = - \frac{χ _{ρ} ( k )}{∣ k ∣ ^{2}} s (k) .

Replacing $N$ in (eq:Poisson) gives $w_{P}_{ρ}$ and the symbol $F^{P}_{κ, ρ}$ .

Displayed formulas in this passage

unlabeled · line 775
$s (k) = (2 π)^{- 3/2} \int_{R^{3}} e^{- i k \cdot x} s (x) d^{3} x .$
unlabeled · line 781
$a = 0 on [0, 1], a = 1 on [2, \infty),$
unlabeled · line 784
$b = 1 on [0, 1], b = 0 on [2, \infty) .$
unlabeled · line 788
$χ_{μ, Λ} (k) = a (∣ k ∣/ μ) b (∣ k ∣/Λ), ρ = (μ, Λ),$
unlabeled · line 794
$N_{ρ} s (k) = - \frac{χ _{ρ} ( k )}{∣ k ∣ ^{2}} s (k) .$