Mean of a function

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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 a one-dimensional domain, 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 ${\displaystyle n}$ 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

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

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