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One-dimensional

In a one-dimensional domain, the mean of a function f(x) over the interval (a,b) is defined by:¹

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

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

∫ a b f ¯ d x = ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}{\bar {f}}\,dx=\int _{a}^{b}f(x)\,dx.}

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

∫ a b f ¯ d x = f ¯ x | a b = f ¯ b − f ¯ a = ( b − a ) f ¯ . {\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 f {\displaystyle f} is continuous then there exists a point c ∈ ( a , b ) {\displaystyle c\in (a,b)} such that

∫ a b f ( x ) d x = f ( c ) ( b − a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=f(c)(b-a).}

The point f ( c ) {\displaystyle f(c)} is called the mean value of f ( x ) {\displaystyle f(x)} on [ a , b ] {\displaystyle [a,b]} . So we write f ¯ = f ( c ) {\displaystyle {\bar {f}}=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

f ¯ = 1 Vol ( U ) ∫ U f d V {\displaystyle {\bar {f}}={\frac {1}{{\hbox{Vol}}(U)}}\int _{U}f\;dV}

where Vol ( U ) {\displaystyle {\hbox{Vol}}(U)} and d V {\displaystyle 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 ⁡ ( 1 Vol ( U ) ∫ U log ⁡ f ) . {\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.

See also

• Mean

References

1. Dougherty, Bradley (2016). "On the Average of a Function and the Mean Value Theorem for Integrals". Pi Mu Epsilon Journal. 14 (4): 251–254. ISSN 0031-952X. JSTOR 48568127. Retrieved 11 January 2023. https://www.jstor.org/stable/48568127 ↩