Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

A compact regularization of the kernels

The named topology and regular approximations › A compact regularization of the kernels · lines 920–946

VII-C@1#eq:Beps

Choose a real, nonnegative, even, Euclidean-radial $ρ \in C_{c}^{\infty} (R^{4})$ with $\int ρ = 1$ , and put

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

Define

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

Its mixed kernel is

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

In particular, $B_{f, ε} \in P_{reg}$ .

These are approximation devices for the exact sharp kernel. They are not proposed as replacement observables.

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