Manuscript Em01-D@2 · Em01-D-v2.tex · theorem · d3b49151

Theorem: Conditional fixed-kernel decision

The fixed-kernel decision problem › Main theorem · lines 273–341

Em01-D@2#thm:main, Em01-D@2#eq:mainexpansion, Em01-D@2#eq:singularsum, Em01-D@2#eq:maincoeff, Em01-D@2#eq:mainfinite, Em01-D@2#eq:mainrem, Em01-D@2#eq:maindrem

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Fix the data of Sections sec:inputs–sec:contraction and a normalized compact profile $ρ$ . Suppose the complete translation distribution has the origin-germ certificate of Assumption ass:germ.

Then the family has the expansion

t_{δ} = S_{δ}^{ρ} + T^{ρ} + R_{δ}^{ρ},

where

S_{δ}^{ρ} = λ < 0 \sum k = 0 \sum^{J_{λ}} δ^{λ} (lo g δ)^{k} A_{λ, k}^{ρ} + k = 1 \sum^{J_{0}} (lo g δ)^{k} A_{0, k}^{ρ} .

Absent exponents contribute no terms. All displayed coefficients are determined by

A_{λ, k}^{ρ} (Φ) T^{ρ} (Φ) = \frac{1}{k !} ⟨ H_{λ, J_{λ} - k} [Φ], ρ ⟩, = W_{Φ} (0) + A_{0, 0}^{ρ} (Φ) .

If $0$ is absent, $A_{0, 0}^{ρ} = 0$ .

For every compact $L ⋐ U$ , there are $m_{L}, C_{L} > 0$ and $a_{L} \in (0, 1]$ such that, for sufficiently small scales and $supp Φ \subset L$ ,

∣ R_{δ}^{ρ} (Φ) ∣ ∣ [δ \partial_{δ} (t_{δ} - S_{δ}^{ρ})] (Φ) ∣ \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}}, \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}} .

Consequently:

the specified raw family converges strongly if and only if every coefficient in (eq:singularsum) is zero;
if one such coefficient is nonzero, one fixed compact joint test detects an unbounded raw pairing;
subtraction of (eq:singularsum) always gives the mathematical finite part $T^{ρ}$ under the certificate, but constitutes an admissible extension only under the additional conditions of Section sec:admissibility.

The proof does not require a continuous characteristic map on the whole internal module.

Displayed formulas in this passage

Em01-D@2#eq:mainexpansion · line 281
$t_{δ} = S_{δ}^{ρ} + T^{ρ} + R_{δ}^{ρ},$
Em01-D@2#eq:singularsum · line 288
$S_{δ}^{ρ} = λ < 0 \sum k = 0 \sum^{J_{λ}} δ^{λ} (lo g δ)^{k} A_{λ, k}^{ρ} + k = 1 \sum^{J_{0}} (lo g δ)^{k} A_{0, k}^{ρ} .$
Em01-D@2#eq:maincoeff · line 299
$A_{λ, k}^{ρ} (Φ) T^{ρ} (Φ) = \frac{1}{k !} ⟨ H_{λ, J_{λ} - k} [Φ], ρ ⟩, = W_{Φ} (0) + A_{0, 0}^{ρ} (Φ) .$
Em01-D@2#eq:mainfinite · line 299
$A_{λ, k}^{ρ} (Φ) T^{ρ} (Φ) = \frac{1}{k !} ⟨ H_{λ, J_{λ} - k} [Φ], ρ ⟩, = W_{Φ} (0) + A_{0, 0}^{ρ} (Φ) .$
Em01-D@2#eq:mainrem · line 315
$∣ R_{δ}^{ρ} (Φ) ∣ ∣ [δ \partial_{δ} (t_{δ} - S_{δ}^{ρ})] (Φ) ∣ \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}}, \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}} .$
Em01-D@2#eq:maindrem · line 315
$∣ R_{δ}^{ρ} (Φ) ∣ ∣ [δ \partial_{δ} (t_{δ} - S_{δ}^{ρ})] (Φ) ∣ \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}}, \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}} .$

Theorem: Conditional fixed-kernel decision

Displayed formulas in this passage

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