Mean of a function

In a one-dimensional domain, the mean ${\displaystyle {\bar {f}}}$ of a function f(x) over the interval [a, b] is defined by^[1] $${\displaystyle {\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.}$$

This definition can be justified as follows. The average value ${\displaystyle {\bar {y}}}$ of finitely many numbers ${\displaystyle y_{1},y_{2},\dots ,y_{n}}$ is defined by the property ${\displaystyle n{\bar {y}}=y_{1}+y_{2}+\cdots +y_{n}}$. In other words, ${\displaystyle {\bar {y}}}$ is the constant value which when added ${\displaystyle n}$ 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 $${\displaystyle \int _{a}^{b}{\bar {f}}\,dx={\bar {f}}x{\bigr |}_{a}^{b}={\bar {f}}b-{\bar {f}}a=(b-a){\bar {f}}.}$$

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

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

where ${\displaystyle {\hbox{Vol}}(U)}$ and ${\displaystyle dV}$ are, respectively, the domain volume and volume element (or generalizations thereof, e.g., volume form).

The above generalizes the arithmetic mean to functions. On the other hand, it is also possible to generalize the geometric mean to functions by:

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

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.

• Mean