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Difference between revisions of "Symmetric derived number"

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''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916201.png" />''
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$#C+1 = 22 : ~/encyclopedia/old_files/data/S091/S.0901620 Symmetric derived number
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A generalization of the ordinary notion of a derived number (cf. [[Dini derivative|Dini derivative]]) to the case of a set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916202.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916203.png" />-dimensional Euclidean space. The symmetric derived numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916204.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916205.png" /> are defined as the limits
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{{TEX|done}}
  
<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/s091/s091620/s0916206.png" /></td> </tr></table>
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''at a point  $  x $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916207.png" /> is some sequence of closed balls with centres at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916208.png" /> and radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s0916209.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162010.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162011.png" />.
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A generalization of the ordinary notion of a derived number (cf. [[Dini derivative|Dini derivative]]) to the case of a set function  $  \Phi $
 +
on an  $  n $-
 +
dimensional Euclidean space. The symmetric derived numbers of $  \Phi $
 +
at $  x $
 +
are defined as the limits
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162013.png" />-th symmetric derived numbers at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162014.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162015.png" /> of a real variable are defined as the limits
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$$
 +
\lim\limits _ {k \rightarrow \infty } \
  
<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/s091/s091620/s09162016.png" /></td> </tr></table>
+
\frac{\Phi ( S ( x, r _ {k} )) }{| S ( x, r _ {k} ) | }
 +
,
 +
$$
  
<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/s091/s091620/s09162017.png" /></td> </tr></table>
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where  $  S ( x, r _ {k} ) $
 +
is some sequence of closed balls with centres at  $  x $
 +
and radii  $  r _ {k} $
 +
such that  $  r _ {k} \rightarrow 0 $
 +
as  $  k \rightarrow \infty $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162018.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162020.png" /> is the [[Symmetric difference of order n|symmetric difference of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162021.png" />]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162022.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091620/s09162023.png" />.
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The  $  n $-
 +
th symmetric derived numbers at  $  x $
 +
of a function  $  f $
 +
of a real variable are defined as the limits
 +
 
 +
$$
 +
\lim\limits _ {k \rightarrow \infty } \
 +
 
 +
\frac{\Delta _ {s}  ^ {n} f ( x, h _ {k} ) }{h _ {k}  ^ {n} }
 +
=
 +
$$
 +
 
 +
$$
 +
= \
 +
\lim\limits _ {k \rightarrow \infty } 
 +
\frac{\sum _ {m = 0 } ^ { n }  \left ( \begin{array}{c}
 +
n \\
 +
m
 +
\end{array}
 +
\right ) (- 1)  ^ {m} f \left
 +
( x + {
 +
\frac{n - 2m }{2}
 +
} h _ {k} \right ) }{h _ {k}  ^ {n} }
 +
,
 +
$$
 +
 
 +
where  $  h _ {k} \rightarrow 0 $
 +
as $  k \rightarrow \infty $
 +
and $  \Delta _ {s}  ^ {n} f ( x, h _ {k} ) $
 +
is the [[Symmetric difference of order n|symmetric difference of order $  n $]]  
 +
of $  f $
 +
at $  x $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1937)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1937)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


at a point $ x $

A generalization of the ordinary notion of a derived number (cf. Dini derivative) to the case of a set function $ \Phi $ on an $ n $- dimensional Euclidean space. The symmetric derived numbers of $ \Phi $ at $ x $ are defined as the limits

$$ \lim\limits _ {k \rightarrow \infty } \ \frac{\Phi ( S ( x, r _ {k} )) }{| S ( x, r _ {k} ) | } , $$

where $ S ( x, r _ {k} ) $ is some sequence of closed balls with centres at $ x $ and radii $ r _ {k} $ such that $ r _ {k} \rightarrow 0 $ as $ k \rightarrow \infty $.

The $ n $- th symmetric derived numbers at $ x $ of a function $ f $ of a real variable are defined as the limits

$$ \lim\limits _ {k \rightarrow \infty } \ \frac{\Delta _ {s} ^ {n} f ( x, h _ {k} ) }{h _ {k} ^ {n} } = $$

$$ = \ \lim\limits _ {k \rightarrow \infty } \frac{\sum _ {m = 0 } ^ { n } \left ( \begin{array}{c} n \\ m \end{array} \right ) (- 1) ^ {m} f \left ( x + { \frac{n - 2m }{2} } h _ {k} \right ) }{h _ {k} ^ {n} } , $$

where $ h _ {k} \rightarrow 0 $ as $ k \rightarrow \infty $ and $ \Delta _ {s} ^ {n} f ( x, h _ {k} ) $ is the symmetric difference of order $ n $ of $ f $ at $ x $.

References

[1] S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French)

Comments

References

[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
How to Cite This Entry:
Symmetric derived number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_derived_number&oldid=17421
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article