Manuscript IX-Free@1 · IX-Free.tex · section · 33b20e96

F1: concrete presentation-relation kernels

F1: concrete presentation-relation kernels · lines 513–545

IX-Free@1#eq:K

The fixed mixed kernel is

K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .

The low-frequency amplitudes are locally integrable in three spatial dimensions. The high-frequency terms are oscillatory distributions.

For $v = \sum_{i} α_{i} f_{i} \otimes g_{i} \in E_{L}$ , write

v (x, y) = i \sum α_{i} f_{i} (x) g_{i} (y) .

This identifies the algebraic tensor product with a subspace of $C_{c}^{\infty} (M_{+}^{2})$ . Injectivity follows by choosing a representation with linearly independent first factors and applying linear functionals on their finite-dimensional span to isolate each second factor.

Displayed formulas in this passage

IX-Free@1#eq:K · line 517
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
unlabeled · line 537
$v (x, y) = i \sum α_{i} f_{i} (x) g_{i} (y) .$