Mean of a function
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by
Recall that a defining property of the average value of finitely many numbers is that . In other words, is the constant value which when added to itself times equals the result of adding the terms of . By analogy, a defining property of the average value of a function over the interval is that
In other words, is the constant value which when integrated over equals the result of integrating over . But by the second fundamental theorem of calculus, the integral of a constant is just
See also the first mean value theorem for integration, which guarantees that if is continuous then there exists a point such that
The point is called the mean value of on . So we write and rearrange the preceding equation to get the above definition.
This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be
More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.
There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.