Distributions

Chapter 8Regularization and Applications

We construct here regularizations, namely sequences of regular approximations, of a distribution f . We construct two types: local, or global.

First of all, we construct local approximations f ρ_n, where ρ_n is a regularizing function, namely regular with support in the ball B(0, 1/n) and whose integral is 1. So, f ρ_n is defined on the open set Ω_1/n obtained by removing from Ω a neighborhood of its boundary of “width” 1/n. It is regular and locally converges to f .

In § 8.3, we construct global approximations where α is a localizing function, namely regular with compact support and equal to 1 except in a neighborhood of the boundary, and where is the extension by 0. So, R_nf is defined on the whole of Ω. It is regular and globally converges to f , at the price of a loss of approximation in the neighborhood of the boundary.

Finally, we establish, by using the regularization:

• — associativity and commutativity properties of weighting, in § 8.6; ...