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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 (for instance, Y could be some set of numbers), then the pointwise product of f and g is another function from X to Y which maps x ∈ X to f(x)g(x).
- given by
be well-defined. Let f and g be functions f, g : X → Y. Then the pointwise product (f ⋅ g) : X → Y is defined by
for each x in X. In the same manner in which the binary operator ⋅ is omitted from products, we say that f ⋅ g = fg.
- If Y is the set of real numbers R, then the pointwise product of f, g : X → R is just normal multiplication of the images. For example, if we have f(x) = 2x and g(x) = x + 1 then
- The convolution theorem states that the Fourier transform of a convolution is the pointwise product of Fourier transforms:
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.
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:
|T||T||T||0.1||0.2||0.1 × 0.2|
|T||T||F||0.1||0.4||0.1 × 0.4|
|T||F||T||0.3||0.6||0.3 × 0.6|
|T||F||F||0.3||0.8||0.3 × 0.8|
|F||T||T||0.5||0.2||0.5 × 0.2|
|F||T||F||0.5||0.4||0.5 × 0.4|
|F||F||T||0.7||0.6||0.7 × 0.6|
|F||F||F||0.7||0.8||0.7 × 0.8|