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:[1]

${\displaystyle {\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.}$

Recall that a defining property of the average value ${\displaystyle {\bar {y}}}$ of finitely many numbers ${\displaystyle y_{1},y_{2},\dots ,y_{n}}$ is that ${\displaystyle n{\bar {y}}=y_{1}+y_{2}+\cdots +y_{n}}$. In other words, ${\displaystyle {\bar {y}}}$ is the constant value which when added to itself ${\displaystyle n}$ times equals the result of adding the ${\displaystyle n}$ terms ${\displaystyle y_{1},\dots ,y_{n}}$. By analogy, a defining property of the average value ${\displaystyle {\bar {f}}}$ of a function over the interval ${\displaystyle [a,b]}$ is that

${\displaystyle \int _{a}^{b}{\bar {f}}\,dx=\int _{a}^{b}f(x)\,dx}$

In other words, ${\displaystyle {\bar {f}}}$ is the constant value which when integrated over ${\displaystyle [a,b]}$ equals the result of integrating ${\displaystyle f(x)}$ over ${\displaystyle [a,b]}$. But the integral of a constant ${\displaystyle {\bar {f}}}$ is just

${\displaystyle \int _{a}^{b}{\bar {f}}\,dx={\bar {f}}x{\bigr |}_{a}^{b}={\bar {f}}b-{\bar {f}}a=(b-a){\bar {f}}}$

See also the first mean value theorem for integration, which guarantees that if ${\displaystyle f}$ is continuous then there exists a point ${\displaystyle c\in (a,b)}$ such that

${\displaystyle \int _{a}^{b}f(x)\,dx=f(c)(b-a)}$

The point ${\displaystyle f(c)}$ is called the mean value of ${\displaystyle f(x)}$ on ${\displaystyle [a,b]}$. So we write ${\displaystyle {\bar {f}}=f(c)}$ and rearrange the preceding equation to get the above definition.

In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

${\displaystyle {\bar {f}}={\frac {1}{{\hbox{Vol}}(U)}}\int _{U}f.}$

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

${\displaystyle \exp \left({\frac {1}{{\hbox{Vol}}(U)}}\int _{U}\log f\right).}$

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.