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{{For|entrywise product|Matrix multiplication#Hadamard product}}
{{Unreferenced|date=December 2009}}
The '''pointwise product''' of two [[function (mathematics)|function]]s is another function, obtained by multiplying the [[Image (mathematics)|image]] of the two functions at each value in the domain. If ''f'' and ''g'' are both functions with [[domain (mathematics)|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'').

==Formal definition==
Let ''X'' and ''Y'' be [[set (mathematics)|set]]s, and let [[multiplication]] be defined in ''Y''&mdash;that is, for each ''y'' and ''z'' in ''Y'' let the [[Product (mathematics)|product]]
: <math>\cdot : Y \times Y \longrightarrow Y</math> given by <math>y \cdot z = yz</math>
be well-defined. Let ''f'' and ''g'' be [[function (mathematics)|function]]s ''f'', ''g'' : ''X'' → ''Y''. Then the '''pointwise product''' (''f'' • ''g'') : ''X'' → ''Y'' is defined by

: <math>(f \cdot g)(x) = f(x) \cdot g(x)</math>

for each ''x'' in ''X''. In the same manner in which the [[binary operator]] • is omitted from products, we say that ''f'' • ''g'' = ''fg''.

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

* If ''Y'' is the set of [[real number]]s '''R''', then the pointwise product of ''f'', ''g'' : ''X'' → '''R''' is just normal multiplication of the images. For example, if we have ''f''(''x'') = 2''x'' and ''g''(''x'') = ''x'' + 1 then

: <math>(fg)(x) = f(x)g(x) = 2x(x + 1) = 2x^2 + 2x\,</math>

for every real number ''x'' in '''R'''.

* The [[convolution theorem]] states that the Fourier transform of a convolution is the pointwise product of Fourier transforms:

: <math>\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}</math>

==Algebraic application of pointwise products==
Let ''X'' be a [[Set (mathematics)|set]] and let ''R'' be a [[ring (mathematics)|ring]]. Since [[addition]] and [[multiplication]] are defined in ''R'', we can construct an algebraic structure known as an [[K-algebra|algebra]]  out of the functions from ''X'' to ''R'' by defining addition, multiplication, and scalar multiplication of functions to be done pointwise. 

If ''R''<sup>&thinsp;''X''</sup> denotes the set of functions from ''X'' to ''R'', then we say that if ''f'', ''g'' are elements of ''R''<sup>&thinsp;''X''</sup>, then ''f'' + ''g'', ''fg'', and ''rf'', the last of which is defined by
: <math>(rf)(x) = rf(x)\,</math>
for all ''r'' in ''R'', are all elements of ''R''<sup>&thinsp;''X''</sup>.

==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 (set theory)|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 ''f''<sub>1</sub>() for the boolean variables ''p'' and ''q'', and ''f''<sub>2</sub>() for the boolean variables ''q'' and ''r'', both with the [[range (mathematics)|range]] in '''R''', the pointwise product of ''f''<sub>1</sub>() and ''f''<sub>2</sub>() is shown in the next table:

<center>
 {| class="wikitable"
 |- style="background:silver"
 | ''p'' || ''q'' || ''r'' || ''f''<sub>1</sub>(''p'', ''q'') || ''f''<sub>2</sub>(''q'', ''r'') || pointwise product 
 |- align="center"
 | T || T || T ||         0.1        ||       0.2          ||       0.1 x 0.2   
 |- align="center"
 | T || T || F ||         0.1        ||       0.4          ||       0.1 x 0.4   
 |- align="center"
 | T || F || T ||         0.3        ||       0.6          ||       0.3 x 0.6   
 |- align="center"
 | T || F || F ||         0.3        ||       0.8          ||       0.3 x 0.8   
 |- align="center"
 | F || T || T ||         0.5        ||       0.2          ||       0.5 x 0.2   
 |- align="center"
 | F || T || F ||         0.5        ||       0.4          ||       0.5 x 0.4   
 |- align="center"
 | F || F || T ||         0.7        ||       0.6          ||       0.7 x 0.6   
 |- align="center"
 | F || F || F ||         0.7        ||       0.8          ||       0.7 x 0.8   
 |}
</center>

==See also==
*[[Pointwise]]

{{DEFAULTSORT:Pointwise Product}}
[[Category:Elementary algebra]]
[[Category:Binary operations]]