Convolution of Measures

Summary

24.1 We define convolution of measures. Two bounded measures are always convolvable (Proposition 24.1.2).

24.2 We supply some necessary and sufficient conditions for a measure and a function to be convolvable (Propositions 24.2.2 and 24.2.3).

24.3 If μ is a measure on G (respectively, a measure with compact support) and if f is a continuous function on G with compact support (respectively, a continuous function), then μ and f are convolvable.

24.4 For every bounded measure μ on G and for every function f in ℒp̄(G) (with 1 ≤ p ≤ +∞ and f are convolvable (Theorems 24.4.1 and 24.4.2).

24.5 For 1 ≤ p ≤ +∞, the endomorphism g ↦ μ̆ * g of A ^P_C (G) is the transpose of f ↦ μ * f (Proposition 24.5.2).

24.6 We study convolution of functions on G.

24.7 We enunciate a few results on regularization of functions (Theorems 24.7.1 and 24.7.2).

24.8 We define Gelfand pairs (important in harmonic analysis). We show that (SO(n + 1, R), SO(n, R)) and (SH(n + 1, R), SO(n, R)) are Gelfand pairs.

In this chapter, all locally compact spaces are supposed to be Hausdorff.