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| | Finding the [[Strong derivative|strong derivative]] of an indefinite integral | | Finding the [[Strong derivative|strong derivative]] of an indefinite integral |
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| − | <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>
| + | $$F(I)=\int\limits_If(x)dx$$ |
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| − | 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 | + | 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 |
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| − | <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>
| + | $$|f|(\ln(1+|f|))^{n-1}$$ |
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| − | 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 | + | 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 |
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| − | <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>
| + | $$\phi(u)=o(u\ln^{n-1}u)$$ |
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| − | 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. | + | 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. |
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| | ====References==== | | ====References==== |
Revision as of 14:11, 30 August 2014
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) |
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