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The expression
 
The expression
  
<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/d030270/d0302701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
V _ {n, p }  ( f, x) = \
  
<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/d030270/d0302702.png" /></td> </tr></table>
+
\frac{1}{p + 1 }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302704.png" /> are the partial sums of the [[Fourier series|Fourier series]] of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302705.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302707.png" />, the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302708.png" />, they become identical with the Fejér sums (cf. [[Fejér sum|Fejér sum]]). Ch.J. de la Vallée-Poussin [[#References|[1]]], [[#References|[2]]] was the first to study the method of approximating periodic functions by polynomials of the form (*); he also established the inequality
+
\sum _ {k = n - p } ^ { n }
 +
S _ {k} ( f, x),
 +
$$
  
<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/d030270/d0302709.png" /></td> </tr></table>
+
$$
 +
p = 0 \dots n; \  n = 0, 1 \dots
 +
$$
  
<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/d030270/d03027010.png" /></td> </tr></table>
+
where  $  S _ {k} ( f, x) $,
 +
$  k = 0, 1 \dots $
 +
are the partial sums of the [[Fourier series|Fourier series]] of a function  $  f $
 +
with period  $  2 \pi $.
 +
If  $  p = 0 $,
 +
the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if  $  p = n $,
 +
they become identical with the Fejér sums (cf. [[Fejér sum|Fejér sum]]). Ch.J. de la Vallée-Poussin [[#References|[1]]], [[#References|[2]]] was the first to study the method of approximating periodic functions by polynomials of the form (*); he also established the inequality
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027011.png" /> is the best uniform approximation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027012.png" /> using trigonometric polynomials of order not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027016.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027017.png" />, the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027018.png" /> realize an approximation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027019.png" />. The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027020.png" /> yield the best order approximations of continuous functions of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027021.png" />, with an estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027022.png" /> for certain values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027024.png" />. The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027027.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027028.png" /> is a trigonometric polynomial of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027030.png" />. A de la Vallée-Poussin sum may be written as follows
+
$$
 +
| f ( x) - V _ {n, p }  ( f, x) |  \leq  \
 +
2
 +
\frac{n + 1 }{p + 1 }
  
<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/d030270/d03027031.png" /></td> </tr></table>
+
E _ {n - p }  ( f  ),
 +
$$
  
<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/d030270/d03027032.png" /></td> </tr></table>
+
$$
 +
= 0 \dots n,
 +
$$
 +
 
 +
where  $  E _ {m} ( f  ) $
 +
is the best uniform approximation of the function  $  f \in C _ {2 \pi }  $
 +
using trigonometric polynomials of order not greater than  $  m $.
 +
If  $  p = [ cn] $,
 +
$  0 < c < 1 $
 +
and  $  [ a] $
 +
is the integer part of the number  $  a $,
 +
the polynomials  $  V _ {n,[ cn] }  ( f, x) $
 +
realize an approximation of order  $  O( E _ {[( 1- c) n] }  ( f  )) $.
 +
The polynomials  $  V _ {n,[ cn] }  ( f, x) $
 +
yield the best order approximations of continuous functions of period  $  2 \pi $,
 +
with an estimate  $  E _ {[ \theta n] }  ( f  ) = O( E _ {n} ( f  )) $
 +
for certain values of  $  \theta $,
 +
0 \leq  \theta < 1 $.  
 +
The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if  $  p = [ cn] $,
 +
$  0 < c < 1 $,
 +
then  $  | V _ {n,p} ( f, x) | \leq  K( c)  \max  | f( x) | $,
 +
and if  $  f $
 +
is a trigonometric polynomial of order not exceeding  $  n - p $,
 +
then  $  V _ {n,p} ( f, x) = f( x) $.
 +
A de la Vallée-Poussin sum may be written as follows
 +
 
 +
$$
 +
V _ {n, p }  ( f, x) =
 +
$$
 +
 
 +
$$
 +
= \
 +
 
 +
\frac{1}{( p + 1) \pi }
 +
\int\limits _ {- \pi } ^  \pi  \left [
 +
f ( x + t)  \sin 
 +
\frac{2n + 1 - p }{2}
 +
 +
\frac{\sin
 +
( p + 1) t / 2 }{2  \sin  ^ {2}  {t / 2 } }
 +
\right ]  dt,
 +
$$
  
 
where the expressions
 
where the expressions
  
<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/d030270/d03027033.png" /></td> </tr></table>
+
$$
 +
K _ {n, p }  ( t)  = \
 +
 
 +
\frac{\sin  (( 2n + 1 - p) t / 2 )  \sin  (( p + 1) t / 2 ) }{2 ( p + 1)  \sin  ^ {2}  {t / 2 } }
 +
,
 +
$$
  
<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/d030270/d03027034.png" /></td> </tr></table>
+
$$
 +
p = 0 \dots n; \  n = 0, 1 \dots
 +
$$
  
 
are said to be the de la Vallée-Poussin kernels.
 
are said to be the de la Vallée-Poussin kernels.
Line 27: Line 102:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur la meilleure approximation des fonctions d'une variable réelle par des expressions d'ordre donné"  ''C.R. Acad. Sci. Paris Sér. I. Math.'' , '''166'''  (1918)  pp. 799–802</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Leçons sur l'approximation des fonctions d'une variable réelle" , Gauthier-Villars  (1919)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1''' , F. Ungar  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,  "Sur certaines méthodes d'approximation au moyen de sommes trigonométriques"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''4''' :  6  (1940)  pp. 509–520</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.B. Stechkin,  "On de la Vallée-Poussin sums"  ''Dokl. Akad. Nauk SSSR'' , '''80''' :  4  (1951)  pp. 545–520  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.D. Shcherbina,  "On a summation method of series, conjugate to Fourier series"  ''Mat. Sb.'' , '''27 (69)''' :  2  (1950)  pp. 157–170  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Timan,  "Approximation properties of linear methods of summation of Fourier series"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''17'''  (1953)  pp. 99–134  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Efimov,  "On approximation of periodic functions by de la Vallée-Poussin sums"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23''' :  5  (1959)  pp. 737–770  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.V. Efimov,  "On approximation of periodic functions by de la Vallée-Poussin sums"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''24''' :  3  (1960)  pp. 431–468  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.A. Telyakovskii,  "Approximation of differentiable functions by de la Vallée-Poussin sums"  ''Dokl. Akad. Nauk SSSR'' , '''121''' :  3  (1958)  pp. 426–429  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  S.A. Telyakovskii,  "Approximation to functions differentiable in Weyl's sense by de la Vallée-Poussin sums"  ''Soviet Math. Dokl.'' , '''1''' :  2  (1960)  pp. 240–243  ''Dokl. Akad. Nauk SSSR'' , '''131''' :  2  (1960)  pp. 259–262</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur la meilleure approximation des fonctions d'une variable réelle par des expressions d'ordre donné"  ''C.R. Acad. Sci. Paris Sér. I. Math.'' , '''166'''  (1918)  pp. 799–802</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Leçons sur l'approximation des fonctions d'une variable réelle" , Gauthier-Villars  (1919)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1''' , F. Ungar  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,  "Sur certaines méthodes d'approximation au moyen de sommes trigonométriques"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''4''' :  6  (1940)  pp. 509–520</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.B. Stechkin,  "On de la Vallée-Poussin sums"  ''Dokl. Akad. Nauk SSSR'' , '''80''' :  4  (1951)  pp. 545–520  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.D. Shcherbina,  "On a summation method of series, conjugate to Fourier series"  ''Mat. Sb.'' , '''27 (69)''' :  2  (1950)  pp. 157–170  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Timan,  "Approximation properties of linear methods of summation of Fourier series"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''17'''  (1953)  pp. 99–134  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Efimov,  "On approximation of periodic functions by de la Vallée-Poussin sums"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23''' :  5  (1959)  pp. 737–770  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.V. Efimov,  "On approximation of periodic functions by de la Vallée-Poussin sums"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''24''' :  3  (1960)  pp. 431–468  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.A. Telyakovskii,  "Approximation of differentiable functions by de la Vallée-Poussin sums"  ''Dokl. Akad. Nauk SSSR'' , '''121''' :  3  (1958)  pp. 426–429  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  S.A. Telyakovskii,  "Approximation to functions differentiable in Weyl's sense by de la Vallée-Poussin sums"  ''Soviet Math. Dokl.'' , '''1''' :  2  (1960)  pp. 240–243  ''Dokl. Akad. Nauk SSSR'' , '''131''' :  2  (1960)  pp. 259–262</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The de la Vallée-Poussin kernels are also given by the following formula, which in a way most clearly reveals their structure:
 
The de la Vallée-Poussin kernels are also given by the following formula, which in a way most clearly reveals their structure:
  
<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/d030270/d03027035.png" /></td> </tr></table>
+
$$
 +
K _ {n,p} ( t)  = \
 +
{
 +
\frac{1}{p + 1 }
 +
}
 +
\sum _ {k = n - p } ^ { n }
 +
D _ {k} ( t) =
 +
$$
  
<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/d030270/d03027036.png" /></td> </tr></table>
+
$$
 +
= \
 +
{
 +
\frac{1}{2}
 +
} + \sum _ {k = 1 } ^ { {n- }  p } \cos  kt
 +
+ \sum _ {k = 1 } ^ { p }  \left ( 1-{
 +
\frac{k}{p + 1 }
 +
} \right ) \
 +
\cos  ( n- p + k) t.
 +
$$
  
Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027038.png" />) are the Dirichlet kernels (cf. [[Dirichlet kernel|Dirichlet kernel]]).
+
Here the $  D _ {k} $(
 +
$  k \geq  0 $)  
 +
are the Dirichlet kernels (cf. [[Dirichlet kernel|Dirichlet kernel]]).

Latest revision as of 17:32, 5 June 2020


The expression

$$ \tag{* } V _ {n, p } ( f, x) = \ \frac{1}{p + 1 } \sum _ {k = n - p } ^ { n } S _ {k} ( f, x), $$

$$ p = 0 \dots n; \ n = 0, 1 \dots $$

where $ S _ {k} ( f, x) $, $ k = 0, 1 \dots $ are the partial sums of the Fourier series of a function $ f $ with period $ 2 \pi $. If $ p = 0 $, the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if $ p = n $, they become identical with the Fejér sums (cf. Fejér sum). Ch.J. de la Vallée-Poussin [1], [2] was the first to study the method of approximating periodic functions by polynomials of the form (*); he also established the inequality

$$ | f ( x) - V _ {n, p } ( f, x) | \leq \ 2 \frac{n + 1 }{p + 1 } E _ {n - p } ( f ), $$

$$ p = 0 \dots n, $$

where $ E _ {m} ( f ) $ is the best uniform approximation of the function $ f \in C _ {2 \pi } $ using trigonometric polynomials of order not greater than $ m $. If $ p = [ cn] $, $ 0 < c < 1 $ and $ [ a] $ is the integer part of the number $ a $, the polynomials $ V _ {n,[ cn] } ( f, x) $ realize an approximation of order $ O( E _ {[( 1- c) n] } ( f )) $. The polynomials $ V _ {n,[ cn] } ( f, x) $ yield the best order approximations of continuous functions of period $ 2 \pi $, with an estimate $ E _ {[ \theta n] } ( f ) = O( E _ {n} ( f )) $ for certain values of $ \theta $, $ 0 \leq \theta < 1 $. The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if $ p = [ cn] $, $ 0 < c < 1 $, then $ | V _ {n,p} ( f, x) | \leq K( c) \max | f( x) | $, and if $ f $ is a trigonometric polynomial of order not exceeding $ n - p $, then $ V _ {n,p} ( f, x) = f( x) $. A de la Vallée-Poussin sum may be written as follows

$$ V _ {n, p } ( f, x) = $$

$$ = \ \frac{1}{( p + 1) \pi } \int\limits _ {- \pi } ^ \pi \left [ f ( x + t) \sin \frac{2n + 1 - p }{2} t \frac{\sin ( p + 1) t / 2 }{2 \sin ^ {2} {t / 2 } } \right ] dt, $$

where the expressions

$$ K _ {n, p } ( t) = \ \frac{\sin (( 2n + 1 - p) t / 2 ) \sin (( p + 1) t / 2 ) }{2 ( p + 1) \sin ^ {2} {t / 2 } } , $$

$$ p = 0 \dots n; \ n = 0, 1 \dots $$

are said to be the de la Vallée-Poussin kernels.

References

[1] Ch.J. de la Vallée-Poussin, "Sur la meilleure approximation des fonctions d'une variable réelle par des expressions d'ordre donné" C.R. Acad. Sci. Paris Sér. I. Math. , 166 (1918) pp. 799–802
[2] Ch.J. de la Vallée-Poussin, "Leçons sur l'approximation des fonctions d'une variable réelle" , Gauthier-Villars (1919)
[3] I.P. Natanson, "Constructive function theory" , 1 , F. Ungar (1964) (Translated from Russian)
[4] P.P. Korovkin, "Linear operators and approximation theory" , Hindushtan Publ. Comp. (1960) (Translated from Russian)
[5] S.M. Nikol'skii, "Sur certaines méthodes d'approximation au moyen de sommes trigonométriques" Izv. Akad. Nauk SSSR Ser. Mat. , 4 : 6 (1940) pp. 509–520
[6] S.B. Stechkin, "On de la Vallée-Poussin sums" Dokl. Akad. Nauk SSSR , 80 : 4 (1951) pp. 545–520 (In Russian)
[7] A.D. Shcherbina, "On a summation method of series, conjugate to Fourier series" Mat. Sb. , 27 (69) : 2 (1950) pp. 157–170 (In Russian)
[8] A.F. Timan, "Approximation properties of linear methods of summation of Fourier series" Izv. Akad. Nauk SSSR Ser. Mat. , 17 (1953) pp. 99–134 (In Russian)
[9] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[10] A.V. Efimov, "On approximation of periodic functions by de la Vallée-Poussin sums" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 5 (1959) pp. 737–770 (In Russian)
[11] A.V. Efimov, "On approximation of periodic functions by de la Vallée-Poussin sums" Izv. Akad. Nauk SSSR Ser. Mat. , 24 : 3 (1960) pp. 431–468 (In Russian)
[12] S.A. Telyakovskii, "Approximation of differentiable functions by de la Vallée-Poussin sums" Dokl. Akad. Nauk SSSR , 121 : 3 (1958) pp. 426–429 (In Russian)
[13] S.A. Telyakovskii, "Approximation to functions differentiable in Weyl's sense by de la Vallée-Poussin sums" Soviet Math. Dokl. , 1 : 2 (1960) pp. 240–243 Dokl. Akad. Nauk SSSR , 131 : 2 (1960) pp. 259–262

Comments

The de la Vallée-Poussin kernels are also given by the following formula, which in a way most clearly reveals their structure:

$$ K _ {n,p} ( t) = \ { \frac{1}{p + 1 } } \sum _ {k = n - p } ^ { n } D _ {k} ( t) = $$

$$ = \ { \frac{1}{2} } + \sum _ {k = 1 } ^ { {n- } p } \cos kt + \sum _ {k = 1 } ^ { p } \left ( 1-{ \frac{k}{p + 1 } } \right ) \ \cos ( n- p + k) t. $$

Here the $ D _ {k} $( $ k \geq 0 $) are the Dirichlet kernels (cf. Dirichlet kernel).

How to Cite This Entry:
De la Vallée-Poussin sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_sum&oldid=12479
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article