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''generalized symmetric derivative''
 
''generalized symmetric derivative''
  
A derivative defined by Ch.J. de la Vallée-Poussin [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302401.png" /> be an even number and let there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302402.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302403.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302404.png" />,
+
A derivative defined by Ch.J. de la Vallée-Poussin [[#References|[1]]]. Let $  r $
 +
be an even number and let there exist a $  \delta > 0 $
 +
such that for all $  t $
 +
with  $  | t | < \delta $,
 +
 
 +
$$ \tag{* }
 +
{
 +
\frac{1}{2}
 +
} \{ f ( x _ {0} + t) + f ( x _ {0} - t) \} =
 +
$$
  
<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/d/d030/d030240/d0302405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
= \
 +
\beta _ {0} +
 +
\frac{t  ^ {2} \beta _ {2} }{2 }
 +
+ \dots +
  
<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/d/d030/d030240/d0302406.png" /></td> </tr></table>
+
\frac{t  ^ {r} \beta _ {r} }{r! }
 +
+ \gamma ( t) t  ^ {r} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302407.png" /> are constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302408.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d0302409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024010.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024011.png" /> is called the de la Vallée-Poussin derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024012.png" />, or the symmetric derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024013.png" />, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024014.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024015.png" />.
+
where $  \beta _ {0} \dots \beta _ {r} $
 +
are constants, $  \gamma ( t) \rightarrow 0 $
 +
as $  t \rightarrow 0 $
 +
and $  \gamma ( 0) = 0 $.  
 +
The number $  \beta _ {r} = f _ {(} r) ( x _ {0} ) $
 +
is called the de la Vallée-Poussin derivative of order $  r $,  
 +
or the symmetric derivative of order $  r $,  
 +
of the function $  f $
 +
at the point $  x _ {0} $.
  
The de la Vallée-Poussin derivatives of odd orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024016.png" /> are defined in a similar manner, equation (*) being replaced by
+
The de la Vallée-Poussin derivatives of odd orders $  r $
 +
are defined in a similar manner, equation (*) being replaced by
  
<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/d/d030/d030240/d03024017.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{2}
 +
} \{ f ( x _ {0} + t) - f ( x _ {0} - t) \} =
 +
$$
  
<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/d/d030/d030240/d03024018.png" /></td> </tr></table>
+
$$
 +
= \
 +
t \beta _ {1} +
 +
\frac{t  ^ {3} \beta _ {3} }{
 +
3! }
 +
+ \dots +
 +
\frac{t  ^ {r} \beta _ {r} }{r! }
 +
+ \gamma ( t) t  ^ {r} .
 +
$$
  
The de la Vallée-Poussin derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024019.png" /> is identical with Riemann's second derivative, often called the Schwarzian derivative. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024020.png" /> exists, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024022.png" />, also exist, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024023.png" /> need not exist. If there exists a finite ordinary two-sided derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024025.png" />. For the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024026.png" />, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024028.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024030.png" /> do not exist. If there exists a de la Vallée-Poussin derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024031.png" />, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024032.png" /> obtained from the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024033.png" /> by term-by-term differentiation repeated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024034.png" /> times is summable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024036.png" /> by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030240/d03024038.png" />, [[#References|[2]]] (cf. [[Cesàro summation methods|Cesàro summation methods]]).
+
The de la Vallée-Poussin derivative $  f _ {(} 2) ( x _ {0} ) $
 +
is identical with Riemann's second derivative, often called the Schwarzian derivative. If $  f _ {(} r) ( x _ {0} ) $
 +
exists, $  f _ {(} r- 2) ( x _ {0} ) $,  
 +
$  r \geq  2 $,  
 +
also exist, but $  f _ {(} r- 1) ( x _ {0} ) $
 +
need not exist. If there exists a finite ordinary two-sided derivative $  f ^ { ( r) } ( x _ {0} ) $,  
 +
then $  f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $.  
 +
For the function $  f( x) = { \mathop{\rm sgn} }  x $,  
 +
for example, $  f _ {(} 2k) ( 0) = 0 $,
 +
$  k = 1, 2 \dots $
 +
and the $  f _ {(} 2k+ 1) ( 0) $,  
 +
$  k = 0, 1 \dots $
 +
do not exist. If there exists a de la Vallée-Poussin derivative $  f _ {(} r) ( x _ {0} ) $,  
 +
the series $  S ^ { ( r) } ( f  ) $
 +
obtained from the Fourier series of $  f $
 +
by term-by-term differentiation repeated $  r $
 +
times is summable at $  x _ {0} $
 +
to $  f _ {(} r) ( x _ {0} ) $
 +
by the method $  ( C, \alpha ) $
 +
for $  \alpha > r $,  
 +
[[#References|[2]]] (cf. [[Cesàro summation methods|Cesàro summation methods]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur l'approximation des fonctions d'une variable reélle et de leurs dériveés par des polynômes et des suites limiteés de Fourier"  ''Bull. Acad. Belg.'' , '''3'''  (1908)  pp. 193–254</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  pp. Chapt.11</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur l'approximation des fonctions d'une variable reélle et de leurs dériveés par des polynômes et des suites limiteés de Fourier"  ''Bull. Acad. Belg.'' , '''3'''  (1908)  pp. 193–254</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  pp. Chapt.11</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


generalized symmetric derivative

A derivative defined by Ch.J. de la Vallée-Poussin [1]. Let $ r $ be an even number and let there exist a $ \delta > 0 $ such that for all $ t $ with $ | t | < \delta $,

$$ \tag{* } { \frac{1}{2} } \{ f ( x _ {0} + t) + f ( x _ {0} - t) \} = $$

$$ = \ \beta _ {0} + \frac{t ^ {2} \beta _ {2} }{2 } + \dots + \frac{t ^ {r} \beta _ {r} }{r! } + \gamma ( t) t ^ {r} , $$

where $ \beta _ {0} \dots \beta _ {r} $ are constants, $ \gamma ( t) \rightarrow 0 $ as $ t \rightarrow 0 $ and $ \gamma ( 0) = 0 $. The number $ \beta _ {r} = f _ {(} r) ( x _ {0} ) $ is called the de la Vallée-Poussin derivative of order $ r $, or the symmetric derivative of order $ r $, of the function $ f $ at the point $ x _ {0} $.

The de la Vallée-Poussin derivatives of odd orders $ r $ are defined in a similar manner, equation (*) being replaced by

$$ { \frac{1}{2} } \{ f ( x _ {0} + t) - f ( x _ {0} - t) \} = $$

$$ = \ t \beta _ {1} + \frac{t ^ {3} \beta _ {3} }{ 3! } + \dots + \frac{t ^ {r} \beta _ {r} }{r! } + \gamma ( t) t ^ {r} . $$

The de la Vallée-Poussin derivative $ f _ {(} 2) ( x _ {0} ) $ is identical with Riemann's second derivative, often called the Schwarzian derivative. If $ f _ {(} r) ( x _ {0} ) $ exists, $ f _ {(} r- 2) ( x _ {0} ) $, $ r \geq 2 $, also exist, but $ f _ {(} r- 1) ( x _ {0} ) $ need not exist. If there exists a finite ordinary two-sided derivative $ f ^ { ( r) } ( x _ {0} ) $, then $ f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $. For the function $ f( x) = { \mathop{\rm sgn} } x $, for example, $ f _ {(} 2k) ( 0) = 0 $, $ k = 1, 2 \dots $ and the $ f _ {(} 2k+ 1) ( 0) $, $ k = 0, 1 \dots $ do not exist. If there exists a de la Vallée-Poussin derivative $ f _ {(} r) ( x _ {0} ) $, the series $ S ^ { ( r) } ( f ) $ obtained from the Fourier series of $ f $ by term-by-term differentiation repeated $ r $ times is summable at $ x _ {0} $ to $ f _ {(} r) ( x _ {0} ) $ by the method $ ( C, \alpha ) $ for $ \alpha > r $, [2] (cf. Cesàro summation methods).

References

[1] Ch.J. de la Vallée-Poussin, "Sur l'approximation des fonctions d'une variable reélle et de leurs dériveés par des polynômes et des suites limiteés de Fourier" Bull. Acad. Belg. , 3 (1908) pp. 193–254
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) pp. Chapt.11
How to Cite This Entry:
De la Vallée-Poussin derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_derivative&oldid=22325
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article