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Line 39: Line 39:
 
====Relation with the integral====
 
====Relation with the integral====
 
As it happens for the ordinary derivative, however, the existence almost everywhere of the approximate derivative does not imply that the fundamental theorem
 
As it happens for the ordinary derivative, however, the existence almost everywhere of the approximate derivative does not imply that the fundamental theorem
of calculus applies, that is: $f$ might be approximately differentiable almost everywhere on the interval $[0,1]$ but nonetheless the identity
+
of calculus applies. More precisely, there are examples of functions $f$ which are classically differentiable almost everywhere on the interval $[0,1]$ but such that the identity
 
\[
 
\[
f (a)-f(b) =\int_a^b f' (t)\, dt
+
f (b)-f(a) =\int_a^b f' (t)\, dt
 
\]
 
\]
might fail on a set of positive measure of extrema $(a,b)\in [0,1]^2$ (cp. with the example in [[Absolute continuity]]).
+
fails on a set of positive measure of extrema $(a,b)\in [0,1]^2$ (cp. with the example in [[Absolute continuity]]).
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/a012850/a01285017.png" /></td> </tr></table>
+
If, on the other hand, $F$ and $f$ are measurable functions on $[0,1]$ taking values in $\mathbb R$ and the identity
 +
\begin{equation}\label{e:fundamental}
 +
F(a)-F(0) =\int_0^a f (t)\, dt
 +
\end{equation}
 +
for a.e. $a\in [0,1]$, then the function $F$ can be redifined on a set of measure zero so that
 +
* $F$ is absolutely continuous
 +
* the identity \ref{e:fundamental} holds for ''every'' $a$
 +
* the function $F$ is almost everywhere differentiable and $F'=f$ almost everywhere
  
is a Denjoy–Khinchin integral, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285018.png" /> almost-everywhere in the interval under consideration (an ordinary derivative need not exist on a set of positive measure). This theorem explains the role played by approximate derivatives in the theory of integrals.
+
====Approximate partial derivatives====
 +
If $E\subset \mathbb R^n$ and $f: E\to\mathbb R^k$ are measurable, approximate partial derivatives can be defined as follows. Consider a system of coordinates
 +
$x_1, \dots , x_n$, a point $p=(p_1, \ldots , p_n)\in E$ and a coordinate $i$. If
 +
*$E':=\{ h\in\mathbb R: (p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$ is measurable
 +
*the map $h\mapsto g(h):= f ((p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$
 +
and the approximate limit
 +
\[
 +
L := {\rm ap} \lim_{h\to p_i} \frac{g(h)- g (a_i)}{h-a_i}
 +
\]
 +
exists and it is finite, then $L$ is the approximate partial derivative of $f$ at $p$ in the $i$-th direction.
  
 +
====Comments====
 
There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval.
 
There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval.
 
Approximate partial derivatives of functions of several real variables are considered as well.
 
 
====Comments====
 
 
The concept of an approximate derivative was introduced by A.Ya. Khinchin in 1916.
 
The concept of an approximate derivative was introduced by A.Ya. Khinchin in 1916.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
+
{|
 
+
|-
 
+
|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)
For other references see [[Approximate limit|Approximate limit]].
+
|-
 +
|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|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan  (1968)
 +
|-
 +
|valign="top"|{{Ref|Ru}}||  W. Rudin, "Principles of mathematical analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}}  {{ZBL|0346.2600}} 
 +
|-
 +
|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 06:50, 7 August 2012

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

A generalization of the concept of a derivative, where the ordinary limit is replaced by an approximate limit.

Definition

Consider a (Lebesgure) measurable set $E\subset \mathbb R$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. If the approximate limit \[ {\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} \] exists at it is finite, then it is called approximate derivative of the function $f$ at $x_0$ (some authors include also the case in which the limit is $\pm\infty$). In that case the function $f$ is called approximately differentiable at $x_0$ and, as a consequence of the definition, $f$ is approximately continuous at $x_0$. The concept can be extended further to functions of several variables: see Approximate differentiability Some authors denote the approximate derivative by $f'_{ap} (x_0)$, whereas some authors keep the notation $f' (x_0)$. Indeed if the classical derivative exists, then it coincides with the approximate derivative, whereas the opposite is false (note, for instance, that if $g$ coincides alomst everywhere with $f$, then $g$ is as well approximately differentiable at $x_0$).

Properties

The following useful proposition relates further the two concept.

Proposition 1 Let $E$, $f$ and $x_0$ be as above. The approximate derivative at $x_0$ exists if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that the classical derivative exists for $f|_F$ at $x_0$. Moreover, the approximate derivative at $x_0$ equals the classical derivative of $f_F$ at the same point.

As a corollary of the previous proposition, the classical rules for the differentiation of a sum, a difference, a product, and a quotient of functions apply to (finite) approximate derivatives as well. The theorem on the differentiation of a compositition of approximately differentiable functions does not apply in general. However it applies to $\varphi\circ f$ if $f$ is approximately differentiable at $x_0$ and $\varphi$ is classically differentiable at $f(x_0)$.

Approximate Dini derivatives and Denjoy-Khinchin theorem

Approximate Dini derivatives are defined by analogy with ordinary Dini derivatives (cf. Dini derivative): $D^+ f (x_0)$, $D^- f (x_0)$ and $d^+ f (x_0)$, $d^- f (x_0)$ are, respectively, the right and left approximate upper limits and the right and left approximate lower limits of the quotient \[ \frac{f(x)-f(x_0)}{x-x_0} \] The following theorem applies.

Theorem 2 (Denjoy-Khinchin) If $E\subset \R$ is (Lebesgue) measurable and $f:E\to \R$ (Lebesgue) measurable, then at almost every point of $x_0\in E$,

  • either $f$ has a finite approximate derivative,
  • or $D^+ f (x_0)= D^- f (x_0)= \infty$,
  • or $d^⁺ f (x_0)= d^- f (x_0)=-\infty$.

Relation with the integral

As it happens for the ordinary derivative, however, the existence almost everywhere of the approximate derivative does not imply that the fundamental theorem of calculus applies. More precisely, there are examples of functions $f$ which are classically differentiable almost everywhere on the interval $[0,1]$ but such that the identity \[ f (b)-f(a) =\int_a^b f' (t)\, dt \] fails on a set of positive measure of extrema $(a,b)\in [0,1]^2$ (cp. with the example in Absolute continuity).

If, on the other hand, $F$ and $f$ are measurable functions on $[0,1]$ taking values in $\mathbb R$ and the identity \begin{equation}\label{e:fundamental} F(a)-F(0) =\int_0^a f (t)\, dt \end{equation} for a.e. $a\in [0,1]$, then the function $F$ can be redifined on a set of measure zero so that

  • $F$ is absolutely continuous
  • the identity \ref{e:fundamental} holds for every $a$
  • the function $F$ is almost everywhere differentiable and $F'=f$ almost everywhere

Approximate partial derivatives

If $E\subset \mathbb R^n$ and $f: E\to\mathbb R^k$ are measurable, approximate partial derivatives can be defined as follows. Consider a system of coordinates $x_1, \dots , x_n$, a point $p=(p_1, \ldots , p_n)\in E$ and a coordinate $i$. If

  • $E':=\{ h\in\mathbb R: (p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$ is measurable
  • the map $h\mapsto g(h):= f ((p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$

and the approximate limit \[ L := {\rm ap} \lim_{h\to p_i} \frac{g(h)- g (a_i)}{h-a_i} \] exists and it is finite, then $L$ is the approximate partial derivative of $f$ at $p$ in the $i$-th direction.

Comments

There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval. The concept of an approximate derivative was introduced by A.Ya. Khinchin in 1916.

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)
[Ro] H.L. Royden, "Real analysis" , Macmillan (1968)
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Sa] S. Saks, "Theory of the integral" , Hafner (1952)
[Th] B.S. Thomson, "Real functions" , Springer (1985)
How to Cite This Entry:
Approximate derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_derivative&oldid=27406
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article