Manuscript IX-E-3@1 · IX-E-3.tex · proposition · 8c41417a

Proposition: Verification of (TC) for the flat realization

A fully specified flat realization and the solid-ball error · lines 665–696 · proof included or linked

The operator (eq:E-flat) satisfies (TC).

Proof. Pair (eq:E-flat) with a compact source $a$ , interchange the compactly supported distributional integrations, and collect the coefficient of $j (v, z)$ . This gives

(E_{0}^{*} a) (v, z) = \int_{t \geq v} \int_{R^{3}} \frac{δ ( t - v - ∣ w - z ∣ )}{4 π ∣ w - z ∣} a (t, w) d^{3} w d t .

At $v = 0$ this is (eq:TC).

Smoothness follows directly by angular integration:

(E_{0}^{*} a) (v, z) = \frac{1}{4 π} \int_{0}^{\infty} r \int_{S^{2}} a (v + r, z + r n) d Ω (n) d r .

On compact sets in $(v, z)$ , the effective $r$ -range is bounded. All derivatives can be passed under the integral. Since $C^{*}$ is first order with constant coefficients, $C^{*} b$ is smooth and locally bounded.

Displayed formulas in this passage

unlabeled · line 673
$(E_{0}^{*} a) (v, z) = \int_{t \geq v} \int_{R^{3}} \frac{δ ( t - v - ∣ w - z ∣ )}{4 π ∣ w - z ∣} a (t, w) d^{3} w d t .$
unlabeled · line 684
$(E_{0}^{*} a) (v, z) = \frac{1}{4 π} \int_{0}^{\infty} r \int_{S^{2}} a (v + r, z + r n) d Ω (n) d r .$

Nearby in this section

section: A fully specified flat realization and the solid-ball error