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A generalization of the concept of differentiability obtained by replacing the ordinary limit by an [[Approximate limit|approximate limit]]. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128601.png" /> of a real variable is called approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128602.png" /> if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128603.png" /> such that
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{{MSC|28A33|49Q15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128604.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
The magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128605.png" /> is called the approximate differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128606.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128607.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128608.png" /> is approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128609.png" /> if and only if it has an [[Approximate derivative|approximate derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286010.png" /> at this point. Approximate differentiability of real functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286011.png" /> real variables is defined in a similar manner. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286013.png" /> is called approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286014.png" /> if
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286015.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286017.png" /> are certain given numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286018.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286019.png" /> is called the approximate differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286020.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286021.png" />.
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====Definition====
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A generalization of the concept of differentiability obtained by replacing the ordinary limit by an [[Approximate limit|approximate limit]]. Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has [[Density of a set|Lebesgue density]] $1$. The map $f$ is approximate differentiable at $x_0$ if there
 +
is a linear map $A:\mathbb R^n\to \mathbb R^k$ such that
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\[
 +
{\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0) - A (x-x_0)}{|x-x_0|} = 0\, .
 +
\]
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$A$ is then called the approximate differential of $f$ at $x_0$. If $n=1$ (i.e. $E$ is a subset of the real line), the map $A$ takes the form $A (t) = a t$: the vector $a$ is then the [[Approximate derivative|approximate derivative]] of $f$ at $x_0$, and it is sometimes denoted by $f'_{ap} (x_0)$.
  
Stepanov's theorem: A real-valued measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286022.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286023.png" /> is approximately differentiable almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286024.png" /> if and only if it has finite approximate partial derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286025.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286026.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286027.png" />; these partial derivatives almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286028.png" /> coincide with the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286030.png" />, respectively, of the approximate differential.
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====Properties====
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If $f$ is approximately differentiable at $x_0$, then it is [[Approximate continuity|approximately continuous]] at $x_0$. The usual rules about uniqueness of the differential, differentiability of sums, products and quotients of functions apply to approximate differentiable functions as well and follow from a useful characterization of approximate differentiability:
  
The concept of approximate differentiability can also be extended to vector functions of one or more real variables.
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'''Proposition 1'''
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Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a  measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. $f$ is approximately differentiable at $x_0$ if and only if there is a measurable set $F$ which has Lebesgue density $1$ at $x_0$ and such that $f|_F$ is classically differentiable at $x_0$. The approximate differentiable of $f$ at $x_0$ coincides then with the classical differential of $f|_F$ at $x_0$.
  
====References====
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The chain rule applies to compositions $\varphi\circ f$ when $f$ is approximately differentiable at $x_0$ and $\varphi$ is '''classically differentiable''' at $f(x_0)$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952) (Translated from French)</TD></TR></table>
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====Stepanov and Federer's Theorems====
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The almost everywhere differentiabiliy of a function can be characterized in the following ways.
 +
 
 +
'''Theorem 2 (Stepanov)'''
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A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if and only if the [[Approximate derivative|approximate partial derivatives]] exist almost everywhere.
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'''Theorem 3 (Federer)'''
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Let $E\subset \mathbb R^n$ be a measurable set with finite measure. A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if for every $\varepsilon > 0$ there is a compact set $F\subset E$ such that $\lambda (E\setminus F)<\varepsilon$ and $f|_F$ is $C^1$ (i.e. there exists an extension $g$ of $f|_F$ to $\mathbb R^n$ which is $C^1$).
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In the latter theorem it follows also that the classical differential of $f$ coincides with the approximate differential at almost every $x_0\in F$.
  
 +
Notable examples of maps which are almost everywhere approximately differentiable are the ones belonging to the [[Sobolev classes (of functions)|Sobolev classes]] $W^{1,p}$ and to the [[Function of bounded variation|BV class]].
  
  
====Comments====
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====References====
For other references see [[Approximate limit|Approximate limit]].
+
{|
 +
|-
 +
|valign="top"|{{Ref|AFP}}||  L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The  Clarendon  Press,  Oxford University Press, New York, 2000.    {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|valign="top"|{{Ref|Br}}|| A.M. Bruckner,  "Differentiation of real functions" , Springer  (1978)
 +
|-
 +
|valign="top"|{{Ref|Fe}}||  H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren  der mathematischen Wissenschaften. Springer-Verlag New York Inc., New  York, 1969.
 +
|-
 +
|valign="top"|{{Ref|Mu}}||  M.E. Munroe,  "Introduction to measure and integration" , Addison-Wesley  (1953)
 +
|-
 +
|valign="top"|{{Ref|Sa}}|| S. Saks,  "Theory of the integral" , Hafner  (1952)
 +
|-
 +
|valign="top"|{{Ref|Th}}|| B.S. Thomson,  "Real functions" , Springer  (1985)
 +
|-
 +
|}

Revision as of 07:18, 6 August 2012

2020 Mathematics Subject Classification: Primary: 28A33 Secondary: 49Q15 [MSN][ZBL]

Definition

A generalization of the concept of differentiability obtained by replacing the ordinary limit by an approximate limit. Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. The map $f$ is approximate differentiable at $x_0$ if there is a linear map $A:\mathbb R^n\to \mathbb R^k$ such that \[ {\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0) - A (x-x_0)}{|x-x_0|} = 0\, . \] $A$ is then called the approximate differential of $f$ at $x_0$. If $n=1$ (i.e. $E$ is a subset of the real line), the map $A$ takes the form $A (t) = a t$: the vector $a$ is then the approximate derivative of $f$ at $x_0$, and it is sometimes denoted by $f'_{ap} (x_0)$.

Properties

If $f$ is approximately differentiable at $x_0$, then it is approximately continuous at $x_0$. The usual rules about uniqueness of the differential, differentiability of sums, products and quotients of functions apply to approximate differentiable functions as well and follow from a useful characterization of approximate differentiability:

Proposition 1 Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. $f$ is approximately differentiable at $x_0$ if and only if there is a measurable set $F$ which has Lebesgue density $1$ at $x_0$ and such that $f|_F$ is classically differentiable at $x_0$. The approximate differentiable of $f$ at $x_0$ coincides then with the classical differential of $f|_F$ at $x_0$.

The chain rule applies to compositions $\varphi\circ f$ when $f$ is approximately differentiable at $x_0$ and $\varphi$ is classically differentiable at $f(x_0)$.

Stepanov and Federer's Theorems

The almost everywhere differentiabiliy of a function can be characterized in the following ways.

Theorem 2 (Stepanov) A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if and only if the approximate partial derivatives exist almost everywhere.

Theorem 3 (Federer) Let $E\subset \mathbb R^n$ be a measurable set with finite measure. A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if for every $\varepsilon > 0$ there is a compact set $F\subset E$ such that $\lambda (E\setminus F)<\varepsilon$ and $f|_F$ is $C^1$ (i.e. there exists an extension $g$ of $f|_F$ to $\mathbb R^n$ which is $C^1$).

In the latter theorem it follows also that the classical differential of $f$ coincides with the approximate differential at almost every $x_0\in F$.

Notable examples of maps which are almost everywhere approximately differentiable are the ones belonging to the Sobolev classes $W^{1,p}$ and to the BV class.


References

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Br] A.M. Bruckner, "Differentiation of real functions" , Springer (1978)
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969.
[Mu] M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953)
[Sa] S. Saks, "Theory of the integral" , Hafner (1952)
[Th] B.S. Thomson, "Real functions" , Springer (1985)
How to Cite This Entry:
Approximate differentiability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_differentiability&oldid=27392
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article