Pointwise product

Short description: Product of the images of two functions at every value in the domain

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 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 in X to f (x)g(x) in Y.

Formal definition

Let X and Y be sets such that Y has a notion of multiplication — that is, there is a binary operation

[math]\displaystyle{ \cdot : Y \times Y \longrightarrow Y }[/math] given by [math]\displaystyle{ y \cdot z = yz. }[/math]

Then given two functions [math]\displaystyle{ f,g: X \to Y, }[/math] the pointwise product [math]\displaystyle{ (f \cdot g): X \to Y }[/math] is defined by

[math]\displaystyle{ (f \cdot g)(x) = f(x) \cdot g(x) }[/math]

for all x in X. Just as we often omit the symbol for the binary operation ⋅ (i.e. we write yz instead of y ⋅ z), we often write fg for f ⋅ g.

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.

• If Y is the set of real numbers [math]\displaystyle{ \mathbb R }[/math], then the pointwise product of [math]\displaystyle{ f,g: X \to \mathbb R }[/math] is just normal multiplication of the images. For example, if we have [math]\displaystyle{ f(x) = 2x }[/math] and [math]\displaystyle{ g(x) = x+1 }[/math] then

  [math]\displaystyle{ (fg)(x) = f(x)g(x) = 2x(x + 1) = 2x^2 + 2x\, }[/math]

  for every x in [math]\displaystyle{ \mathbb R. }[/math]
• The convolution theorem states that the Fourier transform of a convolution is the pointwise product of Fourier transforms:

  [math]\displaystyle{ \mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}. }[/math]

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

[math]\displaystyle{ (rf)(x) = rf(x)\, }[/math]

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 of the boolean variables p and q, and f₂ of the boolean variables q and r, both with the range in [math]\displaystyle{ \mathbb R, }[/math] the pointwise product of f₁ and f₂ is shown in the next table:

[math]\displaystyle{ \begin{array} {|c|c|c|c|c|c|} p & q & r & f_1(p,q) & f_2(q,r) & \text{Pointwise product} \\ \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{array} }[/math]

See also

• Pointwise