chapter  3
10 Pages

Convolutions

The convolution comes up very often in formulas for solutions of partial differential equations. Let f and g be measurable functions on Rn. Then the convolution f ∗ g of f and g is defined by

(f ∗ g)(x) = ∫ Rn

f(x− y)g(y) dy, x ∈ Rn,

provided that the integral exists. In order to know when the integral exists, it is convenient to introduce some standard classes of functions.