Manuscript IX-A-ST@1 · IX-A-ST.tex · section · 6e99d1b2

Polynomial kernels

The exact sharp response and compact polynomial functionals › Polynomial kernels · lines 422–448

IX-A-ST@1#eq:polynomial

Let

E = S^{2} T^{*} U \oplus (U \times C), E = C^{\infty} (U; E) .

A configuration is $Ψ = (h, ϕ)$ . The compact-distribution polynomial carrier consists of finite sums

F (Ψ) = n = 0 \sum^{N} \frac{1}{n !} ⟨ u_{n}, Ψ^{\otimes n} ⟩, u_{n} \in E^{'} (U^{n}; (E^{⊠ n})^{*})_{sym} .

The degree-zero coefficient is a complex number. A compact distribution acts on a smooth function by inserting any cutoff equal to one near its support.

Conjugation means the coefficient involution

F^{†} (Ψ) = F (Ψ) .

On real configurations this is ordinary complex conjugation. This convention preserves complex polynomial dependence.

Displayed formulas in this passage

IX-A-ST@1#eq:polynomial · line 431
$F (Ψ) = n = 0 \sum^{N} \frac{1}{n !} ⟨ u_{n}, Ψ^{\otimes n} ⟩, u_{n} \in E^{'} (U^{n}; (E^{⊠ n})^{*})_{sym} .$
unlabeled · line 425
$E = S^{2} T^{*} U \oplus (U \times C), E = C^{\infty} (U; E) .$
unlabeled · line 443
$F^{†} (Ψ) = F (Ψ) .$

Polynomial kernels

Displayed formulas in this passage

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