Manuscript Oai365-A@1 · Oai365-A.tex · theorem · a3840e3d

Theorem: Finite linear-amplitude obstruction

General boundary extraction: a conditional theorem and obstructions › Why a fixed finite amplitude interface is insufficient · lines 1003–1017 · proof included or linked

Oai365-A@1#thm:finite-mode

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There do not exist a finite $d$ , a linear map $K : H_{Γ} \to C^{d}$ , and a positive matrix $E$ such that

Q_{g, A}^{Γ} (f) = ⟨ K f, E K f ⟩ for every f \in H_{Γ} .

Proof. The right-hand form has rank at most $d$ ; alternatively its kernel contains the finite-codimensional subspace $ker K$ . This contradicts Lemma lem:infinite-rank.

Displayed formulas in this passage

unlabeled · line 1007
$Q_{g, A}^{Γ} (f) = ⟨ K f, E K f ⟩ for every f \in H_{Γ} .$

Nearby in this section

lemma: Lemma: Infinite rank of the boundary energy
section: Why a fixed finite amplitude interface is insufficient