Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · section · 6d700be9

Direct componentwise finite parts

Strong radial finite parts › Direct componentwise finite parts · lines 562–586

I-B-VP-1@1#eq:moments

The following elementary extension lemma will also be used to audit direct cutoff prescriptions.

For $p \in {2, 3, 4}$ and a smooth angular coefficient $A$ , define

f^{\circ}_{A} (x) = r^{- p} A (x / r) .

For a test $f$ , put

M_{k}^{A} [f] = \frac{α}{k !} \int_{R^{1 + 1}} \int_{0}^{2 π} A (ω) (ω \cdot \partial_{x})^{k} f (y, 0) d φ d^{2} y, ω = (cos φ, sin φ) .

When $k < p - 2$ , set

m_{p, k}^{ρ} = \int_{0}^{\infty} ρ (u) u^{k + 1 - p} d u .

Displayed formulas in this passage

I-B-VP-1@1#eq:moments · line 572
$M_{k}^{A} [f] = \frac{α}{k !} \int_{R^{1 + 1}} \int_{0}^{2 π} A (ω) (ω \cdot \partial_{x})^{k} f (y, 0) d φ d^{2} y, ω = (cos φ, sin φ) .$
unlabeled · line 568
$f^{\circ}_{A} (x) = r^{- p} A (x / r) .$
unlabeled · line 582
$m_{p, k}^{ρ} = \int_{0}^{\infty} ρ (u) u^{k + 1 - p} d u .$