Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

Unchanged regular approximants

The compatibility boundary: regular approximation is not admission › Unchanged regular approximants · lines 1243–1263

IX-A@1#eq:approx

Choose a real even compact smooth approximate identity

ρ_{ε} (x) = ε^{- 4} ρ (x / ε), \int ρ = 1, S_{ε} u = ρ_{ε} * u,

and define

B_{f, ε} [h, ϕ] = B_{f} [S_{ε} h, S_{ε} ϕ] .

If $b_{f}$ is the mixed kernel of $B_{f}$ , the approximating mixed kernel is

b_{f, ε} = (ρ_{ε} \otimes ρ_{ε}) * b_{f} .

It is smooth and compact, so $B_{f, ε} \in P_{reg}$ .

Displayed formulas in this passage

IX-A@1#eq:approx · line 1251
$B_{f, ε} [h, ϕ] = B_{f} [S_{ε} h, S_{ε} ϕ] .$
unlabeled · line 1246
$ρ_{ε} (x) = ε^{- 4} ρ (x / ε), \int ρ = 1, S_{ε} u = ρ_{ε} * u,$
unlabeled · line 1259
$b_{f, ε} = (ρ_{ε} \otimes ρ_{ε}) * b_{f} .$