Manuscript Em01-B@3 · Em01-B-v3.tex · theorem · 69189992

Theorem: Summable fixed-kernel test

Reconstruction obstruction and conditional limit tools › A limit for one kernel is weaker than a map theorem · lines 1532–1563 · proof included or linked

Em01-B@3#thm:dyadic-limit

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Fix $K_{*}$ and put $t_{n} = F_{2^{- n}} (K_{*})$ . Suppose on every compact $L ⋐ U$ there are an order $m_{L}$ , summable numbers $a_{n, L}$ , and numbers $b_{n, L} \to 0$ such that

∣ (t_{n + 1} - t_{n}) (Φ) ∣ \leq a_{n, L} ∥Φ ∥_{m_{L}},

and

2^{- n - 1} \leq δ \leq 2^{- n} sup ∣ (F_{δ} (K_{*}) - t_{n}) (Φ) ∣ \leq b_{n, L} ∥Φ ∥_{m_{L}} .

Then the full family has a strong distributional limit, with remainder bounded by

(j \geq n \sum a_{j, L} + b_{n, L}) ∥Φ ∥_{m_{L}}

on that smoothing shell.

Proof. The first estimate makes the telescoping dyadic sequence Cauchy in a fixed finite-order test seminorm. Its limit is a distribution because the initial term and the sum of increments satisfy a common local bound. The second estimate controls every value between dyadic parameters. The displayed tail estimate follows by telescoping and implies strong convergence.

Displayed formulas in this passage

unlabeled · line 1538
$∣ (t_{n + 1} - t_{n}) (Φ) ∣ \leq a_{n, L} ∥Φ ∥_{m_{L}},$
unlabeled · line 1542
$2^{- n - 1} \leq δ \leq 2^{- n} sup ∣ (F_{δ} (K_{*}) - t_{n}) (Φ) ∣ \leq b_{n, L} ∥Φ ∥_{m_{L}} .$
unlabeled · line 1549
$(j \geq n \sum a_{j, L} + b_{n, L}) ∥Φ ∥_{m_{L}}$

Theorem: Summable fixed-kernel test

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