pointwise product

In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be multiplied (for instance, could be some set of numbers), then the pointwise product of and is another function from to which maps in to in .

Formal definition

Let and be sets such that has a notion of multiplication — that is, there is a binary operation
: $\cdot : Y \times Y \longrightarrow Y$ given by $y \cdot z = yz.$
Then given two functions $f,g: X \to Y,$ the pointwise product $(f \cdot g): X \to Y$ is defined by

: $(f \cdot g)(x) = f(x) \cdot g(x)$

for all in . Just as we often omit the symbol for the binary operation ⋅ (i.e. we write instead of ), we often write for .

Examples

The most common case of the pointwise product of two functions is when the codomain is a ring (or field), in which multiplication is well-defined.

Algebraic application of pointwise products

Let be a set and let be a ring. Since addition and multiplication are defined in , we can construct an algebraic structure known as an algebra out of the functions from to by defining addition, multiplication, and scalar multiplication of functions to be done pointwise.

If denotes the set of functions from to , then we say that if are elements of , then , , and — the last of which is defined by
: $(rf)(x) = rf(x)\,$

for all in — are all elements of .

Generalization

If both and have as their domain all possible assignments of a set of discrete variables, then their pointwise product is a function whose domain is constructed by all possible assignments of the union of both sets. The value of each assignment is calculated as the product of the values of both functions given to each one the subset of the assignment that is in its domain.

For example, given the function of the boolean variables and , and of the boolean variables and , both with the range in the pointwise product of and is shown in the next table:

:$\begin p & q & r & f_1(p,q) & f_2(q,r) & \text \\ \hline \rm T & \rm T & \rm T & 0.1 & 0.2 & 0.1 \times 0.2 \\ \rm T & \rm T & \rm F & 0.1 & 0.4 & 0.1 \times 0.4 \\ \rm T & \rm F & \rm T & 0.3 & 0.6 & 0.3 \times 0.6 \\ \rm T & \rm F & \rm F & 0.3 & 0.8 & 0.3 \times 0.8 \\ \rm F & \rm T & \rm T & 0.5 & 0.2 & 0.5 \times 0.2 \\ \rm F & \rm T & \rm F & 0.5 & 0.4 & 0.5 \times 0.4 \\ \rm F & \rm F & \rm T & 0.7 & 0.6 & 0.7 \times 0.6 \\ \rm F & \rm F & \rm F & 0.7 & 0.8 & 0.7 \times 0.8 \\ \end$

See also

*Pointwise

{{DEFAULTSORT:Pointwise Product
Elementary algebra
Binary operations