Mechanics
The terminology weight function arises from mechanics: if one has a collection of objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations , then the lever will be in balance if the fulcrum of the lever is at the center of mass
which is also the weighted average of the positions .
In the continuous setting, a weight is a positive measure such as on some domain , which is typically a subset of a Euclidean space , for instance could be an interval . Here is Lebesgue measure and is a non-negative measurable function. In this context, the weight function is sometimes referred to as a density.
Weighted volume
If E is a subset of , then the volume vol(E) of E can be generalized to the weighted volume
Weighted average
If has finite non-zero weighted volume, then we can replace the unweighted average
by the weighted average
If and are two functions, one can generalize the unweighted bilinear form
- :=\int _{\Omega }f(x)g(x)\ dx}
to a weighted bilinear form
- :=\int _{\Omega }f(x)g(x)\ w(x)\ dx.}
See the entry on orthogonal polynomials for examples of weighted orthogonal functions.