mean of a function

In a one-dimensional domain, the mean of a function f(x) over the interval (a,b) is defined by:{{cite journal |last1=Dougherty |first1=Bradley |title=On the Average of a Function and the Mean Value Theorem for Integrals |journal=Pi Mu Epsilon Journal |date=2016 |volume=14 |issue=4 |pages=251–254 |jstor=48568127 |url=https://www.jstor.org/stable/48568127 |access-date=11 January 2023 |issn=0031-952X}}

: $\backslash bar\{f\}=\backslash frac\{1\}\{b-a\}\backslash int\_a^bf(x)\backslash ,dx.$

Recall that a defining property of the average value $\backslash bar\{y\}$ of finitely many numbers $y\_1,\; y\_2,\; \backslash dots,\; y\_n$

is that $n\backslash bar\{y\}\; =\; y\_1\; +\; y\_2\; +\; \backslash cdots\; +\; y\_n$. In other words, $\backslash bar\{y\}$ is the constant value which when

added $n$ times equals the result of adding the $n$ terms $y\_1,\; \backslash dots,\; y\_n$. By analogy, a

defining property of the average value $\backslash bar\{f\}$ of a function over the interval $[a,b]$ is that

: $\backslash int\_a^b\backslash bar\{f\}\backslash ,dx\; =\; \backslash int\_a^bf(x)\backslash ,dx\; .$

In other words, $\backslash bar\{f\}$ is the constant value which when integrated over $[a,b]$ equals the result of

integrating $f(x)$ over $[a,b]$. But the integral of a constant $\backslash bar\{f\}$ is just

: $\backslash int\_a^b\backslash bar\{f\}\backslash ,dx\; =\; \backslash bar\{f\}x\backslash bigr|\_a^b\; =\; \backslash bar\{f\}b\; -\; \backslash bar\{f\}a\; =\; (b\; -\; a)\backslash bar\{f\}\; .$

See also the first mean value theorem for integration, which guarantees

that if $f$ is continuous then there exists a point $c\; \backslash in\; (a,\; b)$ such that

: $\backslash int\_a^bf(x)\backslash ,dx\; =\; f(c)(b\; -\; a)\; .$

The point $f(c)$ is called the mean value of $f(x)$ on $[a,b]$. So we write

$\backslash bar\{f\}\; =\; f(c)$ 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

:$\backslash bar\{f\}=\backslash frac\{1\}\{\backslash hbox\{Vol\}(U)\}\backslash int\_U\; f\; \backslash ;\; dV$

where $\backslash hbox\{Vol\}(U)$ and $dV$ 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:

:$\backslash exp\backslash left(\backslash frac\{1\}\{\backslash hbox\{Vol\}(U)\}\backslash int\_U\; \backslash log\; f\backslash right).$

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

Category:Means

Category:Calculus

{{reflist}}