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Finding the [[Strong derivative|strong derivative]] of an indefinite integral
 
Finding the [[Strong derivative|strong derivative]] of an indefinite integral
  
<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/s/s090/s090530/s0905301.png" /></td> </tr></table>
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$$F(I)=\int\limits_If(x)\,dx$$
  
of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905302.png" /> that is summable in an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905303.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905304.png" />-dimensional Euclidean space, considered as a function of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905305.png" />. If
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of a real-valued function $f$ that is summable in an open subset $G$ of $n$-dimensional Euclidean space, considered as a function of the interval $I\subset G$. 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/s/s090/s090530/s0905306.png" /></td> </tr></table>
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$$|f|(\ln(1+|f|))^{n-1}$$
  
is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905307.png" /> (in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s0905309.png" />), then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053011.png" /> is strongly differentiable almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053012.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053014.png" />, that is positive, non-decreasing and such that
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is summable on $G$ (in particular, if $f\in L_p(G)$, $p>1$), then the integral $F$ of $f$ is strongly differentiable almost-everywhere on $G$. For any $\phi(u)$, $u\geq0$, that is positive, non-decreasing and such that
  
<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/s/s090/s090530/s09053015.png" /></td> </tr></table>
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$$\phi(u)=o(u\ln^{n-1}u)$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053016.png" />, there is a summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053019.png" /> is also summable and such that the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053020.png" /> is unbounded at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053021.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053022.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053023.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090530/s09053024.png" /> cannot be strongly differentiated.
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as $u\to\infty$, there is a summable function $f\geq0$ on $G$ such that $\phi\circ f$ is also summable and such that the ratio $F(I)/|I|$ is unbounded at each $x\in G$, as $I$ tends to $x$, that is, $F$ cannot be strongly differentiated.
  
 
====References====
 
====References====

Latest revision as of 14:20, 14 February 2020

Finding the strong derivative of an indefinite integral

$$F(I)=\int\limits_If(x)\,dx$$

of a real-valued function $f$ that is summable in an open subset $G$ of $n$-dimensional Euclidean space, considered as a function of the interval $I\subset G$. If

$$|f|(\ln(1+|f|))^{n-1}$$

is summable on $G$ (in particular, if $f\in L_p(G)$, $p>1$), then the integral $F$ of $f$ is strongly differentiable almost-everywhere on $G$. For any $\phi(u)$, $u\geq0$, that is positive, non-decreasing and such that

$$\phi(u)=o(u\ln^{n-1}u)$$

as $u\to\infty$, there is a summable function $f\geq0$ on $G$ such that $\phi\circ f$ is also summable and such that the ratio $F(I)/|I|$ is unbounded at each $x\in G$, as $I$ tends to $x$, that is, $F$ cannot be strongly differentiated.

References

[1] B. Jessen, J. Marcinkiewicz, A. Zygmund, "Note on the differentiability of multiple integrals" Fund. Math. , 25 (1935) pp. 217–234
[2] S. Saks, "On the strong derivatives of functions of intervals" Fund. Math. , 25 (1935) pp. 235–252
[3] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[4] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)


Comments

References

[a1] A. Zygmund, "On the differentiability of multiple integrals" Fund. Math. , 23 (1934) pp. 143–149
How to Cite This Entry:
Strong differentiation of an indefinite integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_differentiation_of_an_indefinite_integral&oldid=11851
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article