Arithmetic 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.

One-dimensional

In a one-dimensional domain, the mean of a function ''f''(''x'') over the interval (''a'',''b'') is defined by:

: $\bar=\frac\int_a^bf(x)\,dx.$

Recall that a defining property of the average value $\bar$ of finitely many numbers $y_1, y_2, \dots, y_n$
is that $n\bar = y_1 + y_2 + \cdots + y_n$. In other words, $\bar$ is the ''constant'' value which when
''added'' $n$ times equals the result of adding the $n$ terms $y_1, \dots, y_n$. By analogy, a
defining property of the average value $\bar$ of a function over the interval ,b/math> is that

: $\int_a^b\bar\,dx = \int_a^bf(x)\,dx .$

In other words, $\bar$ is the ''constant'' value which when '' integrated'' over ,b/math> equals the result of
integrating $f(x)$ over ,b/math>. But the integral of a constant $\bar$ is just

: $\int_a^b\bar\,dx = \barx\bigr, _a^b = \barb - \bara = (b - a)\bar .$

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

: $\int_a^bf(x)\,dx = f(c)(b - a) .$

The point $f(c)$ is called the ''mean value'' of $f(x)$ on ,b/math>. So we write
$\bar = f(c)$ and rearrange the preceding equation to get the above definition.

Multi-dimensional

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

:$\bar=\frac\int_U f \; dV$

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

Non-arithmetic

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

:$\exp\left(\frac\int_U \log f\right).$

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.

See also

*Mean

Means
Calculus

References

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