Manuscript XV-F-patch@1 · XV-F-patch.tex · lemma · 6d6ef16d

Lemma: Coefficientwise linearity

Patch I: the local amplitude module and reconstruction › The coefficientwise action is explicit · lines 1303–1327 · proof included or linked

XV-F-patch@1#lem:linearity, XV-F-patch@1#eq:modulelinearity

For the transcribed local coefficient module,

Rec_{V, m} (h c) = h Rec_{V, m} (c), h \in C^{\infty} (V), c \in P_{V, m} .

Consequently the local reconstruction kernel is a $C^{\infty} (V)$ -submodule.

Proof. Substitute $(h c)_{j β} = h c_{j β}$ in (eq:reconstruction). Associativity of multiplication of a distribution by smooth functions gives

j, β \sum h c_{j β} δ^{(j)} (q) \partial_{r}^{β} R (r) = h j, β \sum c_{j β} δ^{(j)} (q) \partial_{r}^{β} R (r) .

If the second sum is zero, multiplication by $h$ remains zero.

Displayed formulas in this passage

XV-F-patch@1#eq:modulelinearity · line 1306
$Rec_{V, m} (h c) = h Rec_{V, m} (c), h \in C^{\infty} (V), c \in P_{V, m} .$
unlabeled · line 1319
$j, β \sum h c_{j β} δ^{(j)} (q) \partial_{r}^{β} R (r) = h j, β \sum c_{j β} δ^{(j)} (q) \partial_{r}^{β} R (r) .$