Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

A precise class of kernel identities

The Sommerfeld–Carslaw–Dowker kernel › A precise class of kernel identities · lines 1084–1101

I-B-G@2#eq:master

We do not assert a master formula for arbitrary bounded Borel functions. The following theorem concerns the heat semigroup, the positive resolvent, and the retarded wave kernel, with the latter interpreted distributionally.

For any radial flat kernel $F (R)$ for which the following expression is defined, write

S_{β} [F] = k \in K (ϑ) \sum e^{2 π i β k} F (R_{k}) - \frac{1}{π α} \int_{0}^{\infty} Φ_{β} (u, ϑ) F (ρ (u)) d u .

Displayed formulas in this passage

I-B-G@2#eq:master · line 1093
$S_{β} [F] = k \in K (ϑ) \sum e^{2 π i β k} F (R_{k}) - \frac{1}{π α} \int_{0}^{\infty} Φ_{β} (u, ϑ) F (ρ (u)) d u .$

Nearby in this section

theorem: Theorem: Heat, resolvent, and wave representation
remark: Remark: Why a null-set argument would not suffice