Distributions

Chapter 7Weighting

In this chapter, we construct the weighted distribution f µ of a distribution f ∈ ′(Ω; E) by a weight which has compact support µ ′_D(^d). Its role will be central in the construction of primitives (Chapter 13) and in the separation of variables (Chapter 15).

Weighting plays an analogous role for Ω to that played by the convolution for ^d. But, since f is not necessarily extendable, f µ is only defined on the open set Ω_D = {x : x + D ⊂ Ω}. We call it weighting since, in the case of functions, i.e. the integral of f in the neighborhood of x with the weight μ.

The general definition of (Definition 7.12), uses the case where the weight is regular to make sense of Before ...