Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Four normalized compact Ward pairs

A finite operational encoding of descended evaluations › Four normalized compact Ward pairs · lines 1782–1821

III-B@1#eq:Fk, III-B@1#eq:integer-table

Fix the nonnegative spatial bump $b$ of Section sec:ward, and put

B_{b} = \int b (x) d^{3} x > 0.

For each $k$ , choose nonnegative nonzero $a_{k} \in C_{c}^{\infty} (I_{k})$ , and define

C_{k} = 2 B_{b} \int a_{k} (s) csc^{2} (m s) d s > 0, a_{k} = \frac{k + 1}{C _{k}} a_{k} .

Use (eq:positive-pair) with $a_{0}$ replaced by $a_{k}$ ; call the resulting pair $(J_{k}, q_{k})$ . Then

F_{k} := O_{J_{k}, q_{k}}, F_{k} (0, Φ) = k + 1.

All four functionals descend on $S_{r_{*}}^{red}$ .

Take the finite classical catalogue

z_{j} = [(0, j Φ)], j = 0, \dots, 7.

The classes are distinct because $F_{0} (z_{j}) = j$ . Their table is

F_{k} (z_{j}) = j (k + 1) .

This normalization produces exact integer evaluation data without claiming numerical quadrature. The positive normalization integrals are part of the displayed classical definitions.

Displayed formulas in this passage

III-B@1#eq:Fk · line 1799
$F_{k} := O_{J_{k}, q_{k}}, F_{k} (0, Φ) = k + 1.$
III-B@1#eq:integer-table · line 1813
$F_{k} (z_{j}) = j (k + 1) .$
unlabeled · line 1786
$B_{b} = \int b (x) d^{3} x > 0.$
unlabeled · line 1791
$C_{k} = 2 B_{b} \int a_{k} (s) csc^{2} (m s) d s > 0, a_{k} = \frac{k + 1}{C _{k}} a_{k} .$
unlabeled · line 1809
$z_{j} = [(0, j Φ)], j = 0, \dots, 7.$