Definition:Pointwise Multiplication of Complex-Valued Functions

Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \Z$ be complex-valued functions.

Then the pointwise product of $f$ and $g$ is defined as:

$f \times g: S \to \Z:$

$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the right hand side is complex multiplication.

Thus pointwise multiplication is seen to be an instance of a pointwise operation on complex-valued functions.

Also denoted as

Using the other common notational forms for multiplication, this definition can also be written:

$\forall s \in S: \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$

or:

$\forall s \in S: \map {\paren {f g} } s := \map f s \map g s$

Also see

• Pointwise Multiplication on Complex-Valued Functions is Associative
• Pointwise Multiplication on Complex-Valued Functions is Commutative

• Definition:Pointwise Operation on Complex-Valued Functions