Pointwise product


The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied, then the pointwise product of f and g is another function from X to Y which maps x ∈ X to fg.

Formal definition

Let X and Y be sets, and let multiplication be defined in Y—that is, for each y and z in Y let the product
be well-defined. Let f and g be functions. Then the pointwise product is defined by
for each x in X. In the same manner in which the binary operator ⋅ is omitted from products, we say that.

Examples

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

Algebraic application of pointwise products

Let X be a set and let R be a ring. Since addition and multiplication are defined in R, we can construct an algebraic structure known as an algebra out of the functions from X to R by defining addition, multiplication, and scalar multiplication of functions to be done pointwise.
If R X denotes the set of functions from X to R, then we say that if f, g are elements of R X, then f + g, fg, and rf, the last of which is defined by
for all r in R, are all elements of R X.

Generalization

If both f and g 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 f1 for the boolean variables p and q, and f2 for the boolean variables q and r, both with the range in R, the pointwise product of f1 and f2 is shown in the next table:
pqrPointwise product
TTT0.10.20.1 × 0.2
TTF0.10.40.1 × 0.4
TFT0.30.60.3 × 0.6
TFF0.30.80.3 × 0.8
FTT0.50.20.5 × 0.2
FTF0.50.40.5 × 0.4
FFT0.70.60.7 × 0.6
FFF0.70.80.7 × 0.8