Convolution

Definition of Convolution

Let $\mu$ and $v$ be positive measures. In the context of harmonic analysis on locally compact commutative groups $G$, the convolution $\mu * v$ exists and is also a positive measure on $G$ if for any test function $f \in {C_c}^{+}(G)$ where ${C_c}^{+}(G)$ is the space of continuous functions with compact support and non-negative values on G; the convolution $\mu * v$ given by the mapping

$$f \mapsto \int_{G \times G} f(x + y) d(\mu \otimes v)(x,y) \lt \infty$$

is finite where $\mu \otimes v$ is the product measure on $G \times G$.

Additional Properties

• Associativity: The convolution operation is associative among positive measures.
• Commutativity: $\mu * v = v * \mu$
• Set $D^+(\mu)$: This is the set of all positive measures $v$ for which $\mu * v$ exists.