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

Definition: Presentation and reconstructed spaces

Internal reconstruction and all-region frequency estimates › Defined seeds and finite reconstruction · lines 757–790

Em01-B@3#def:presentation, Em01-B@3#eq:Rec5

For $0 \leq d \leq 4$ , the $K_{5}$ array spaces are

P_{5, \pm, d} = j + ∣ β ∣ \leq d \prod C^{\infty} (W; V_{int}),

with reconstruction

Rec_{5, \pm, d} (c) = j + ∣ β ∣ \leq d \sum c_{\pm, j β} (w) δ^{(j)} (q_{\pm}) \partial_{r_{\pm}}^{β} R (r_{\pm}) .

For $K_{47}$ , use arrays multiplying derivatives of order at most four of $R (b - a) R (b - z)$ . For $K_{6}$ , use arrays multiplying $\partial_{n}^{α} δ^{(5)} (n)$ , $∣ α ∣ \leq 2$ . For $K_{8}, K_{9}$ , use separate such arrays with $∣ α ∣ \leq 3$ .

Their finite direct sum is $P_{int}$ . The reconstruction map $Rec$ sums the blocks, and

K_{int} = im Rec \subset D^{'} (W; V_{int})

carries the quotient topology transported from $P_{int} / ker Rec$ . All array spaces use compact-set $C^{\infty}$ seminorms.

Displayed formulas in this passage

Em01-B@3#eq:Rec5 · line 766
$Rec_{5, \pm, d} (c) = j + ∣ β ∣ \leq d \sum c_{\pm, j β} (w) δ^{(j)} (q_{\pm}) \partial_{r_{\pm}}^{β} R (r_{\pm}) .$
unlabeled · line 760
$P_{5, \pm, d} = j + ∣ β ∣ \leq d \prod C^{\infty} (W; V_{int}),$
unlabeled · line 784
$K_{int} = im Rec \subset D^{'} (W; V_{int})$

Definition: Presentation and reconstructed spaces

Displayed formulas in this passage

Nearby in this section