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The de la Vallée-Poussin theorem on the distribution of prime numbers: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302901.png" /> be the number of primes smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302902.png" />; then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302903.png" />, the following equality is valid:
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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/d/d030/d030290/d0302904.png" /></td> </tr></table>
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The de la Vallée-Poussin theorem on the distribution of prime numbers: Let $\pi(x)$ be the number of primes smaller than $x$; then, if $x \ge 1$, the following equality is valid:
 +
$$
 +
\pi(x) = \mathrm{li}(x) + O\left({ x \exp(-C\sqrt{\log x}) }\right)
 +
$$
 +
where $C$ is a positive constant and $\mathrm{li}(x)$ is the [[logarithmic integral]] of $x$. This theorem demonstrates the correctness of Gauss' hypothesis on the distribution of prime numbers, viz., as $x \rightarrow \infty$,
 +
$$
 +
\pi(x) \sim \frac{x}{\log x} \ .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302905.png" /> is a positive constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302906.png" /> is the [[Integral logarithm|integral logarithm]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302907.png" />. This theorem demonstrates the correctness of Gauss' hypothesis on the distribution of prime numbers, viz., as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d0302908.png" />,
+
Established by Ch.J. de la Vallée-Poussin [[#References|[1]]]. Cf. [[Distribution of prime numbers]].
 
 
<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/d030290/d0302909.png" /></td> </tr></table>
 
 
 
Established by Ch.J. de la Vallée-Poussin [[#References|[1]]]. Cf. [[Distribution of prime numbers|Distribution of prime numbers]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Recherches analytiques sur la théorie des nombers premiers"  ''Ann. Soc. Sci. Bruxelles'' , '''20'''  (1899)  pp. 183–256</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur la fonction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029010.png" /> de Riemann et la nombre des nombres premiers inférieurs à une limite donnée"  ''Mem. Couronnes Acad. Sci. Belg.'' , '''59''' :  1  (1899–1900)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Recherches analytiques sur la théorie des nombers premiers"  ''Ann. Soc. Sci. Bruxelles'' , '''20'''  (1899)  pp. 183–256</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur la fonction $\zeta(s)$ de Riemann et la nombre des nombres premiers inférieurs à une limite donnée"  ''Mem. Couronnes Acad. Sci. Belg.'' , '''59''' :  1  (1899–1900)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR>
 +
</table>
  
 
''S.M. Vorazhin''
 
''S.M. Vorazhin''
  
The de la Vallée-Poussin alternation theorem: If a sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029012.png" />, in a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029013.png" /> forms an alternation, then for the [[Best approximation|best approximation]] of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029014.png" /> by polynomials of the form
+
The de la Vallée-Poussin alternation theorem: If a sequence of points $  \{ x _ {i} \} $,  
 +
$  i = 0 \dots n + 1 $,  
 +
in a closed set $  Q \in [a, b] $
 +
forms an alternation, then for the [[Best approximation|best approximation]] of a function $  f $
 +
by polynomials of the form
  
<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/d030290/d03029015.png" /></td> </tr></table>
+
$$
 +
P _ {n} (x)  = \
 +
\sum _ {k = 0 } ^ { n }
 +
c _ {k} s _ {k} (x),
 +
$$
  
 
the estimate
 
the estimate
  
<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/d030290/d03029016.png" /></td> </tr></table>
+
$$
 +
E _ {n} (f  )  = \
 +
\inf _ {c _ {k} } \
 +
\sup _ {x \in Q } \
 +
\left | f (x) - \sum _ {k = 0 } ^ { n }
 +
c _ {k} s _ {k} (x) \right | \geq
 +
$$
  
<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/d030290/d03029017.png" /></td> </tr></table>
+
$$
 +
\geq  \
 +
\mathop{\rm min} _ {0 \leq  i \leq  n + 1 } \
 +
| f (x _ {i} ) - P _ {n} (x _ {i} ) |
 +
$$
  
is valid, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029018.png" /> is a Chebyshev system. Established by Ch.J. de la Vallée-Poussin [[#References|[1]]].
+
is valid, where $  {\{ s _ {k} (x) \} } _ {0}  ^ {n} $
 +
is a Chebyshev system. Established by Ch.J. de la Vallée-Poussin [[#References|[1]]].
  
According to the [[Chebyshev theorem|Chebyshev theorem]], equality holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029019.png" /> is the polynomial of best approximation. Analogues of this theorem exist for arbitrary Banach spaces [[#References|[2]]]. The theorem is employed in numerical methods for constructing polynomials of best approximation.
+
According to the [[Chebyshev theorem|Chebyshev theorem]], equality holds if and only if $  P _ {n} (x) $
 +
is the polynomial of best approximation. Analogues of this theorem exist for arbitrary Banach spaces [[#References|[2]]]. The theorem is employed in numerical methods for constructing polynomials of best approximation.
  
 
====References====
 
====References====
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An account of the life and work of de la Vallée-Poussin can be found in, e.g., [[#References|[a1]]].
 
An account of the life and work of de la Vallée-Poussin can be found in, e.g., [[#References|[a1]]].
  
A sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029021.png" />, is called an alternation for a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029024.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030290/d03029025.png" />.
+
A sequence of points $  x _ {i} $,
 +
$  a \leq  x _ {1} < \dots < x _ {m} \leq  b $,  
 +
is called an alternation for a continuous function $  g $
 +
on $  [ a , b ] $
 +
if $  g ( x _ {i} ) = ( - 1 )  ^ {i} \| g \| $
 +
where $  \| g \| = \max _ {x \in [ a , b ] }  | g (x) | $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Favard,  "Hommage à Charles de la Vallée Poussin (1866–1962)"  P.L. Butzer (ed.)  J. Korevaar (ed.) , ''On approximation theory'' , Birkhäuser  (1964)  pp. 1–3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Favard,  "Hommage à Charles de la Vallée Poussin (1866–1962)"  P.L. Butzer (ed.)  J. Korevaar (ed.) , ''On approximation theory'' , Birkhäuser  (1964)  pp. 1–3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>

Latest revision as of 14:03, 17 March 2020


The de la Vallée-Poussin theorem on the distribution of prime numbers: Let $\pi(x)$ be the number of primes smaller than $x$; then, if $x \ge 1$, the following equality is valid: $$ \pi(x) = \mathrm{li}(x) + O\left({ x \exp(-C\sqrt{\log x}) }\right) $$ where $C$ is a positive constant and $\mathrm{li}(x)$ is the logarithmic integral of $x$. This theorem demonstrates the correctness of Gauss' hypothesis on the distribution of prime numbers, viz., as $x \rightarrow \infty$, $$ \pi(x) \sim \frac{x}{\log x} \ . $$

Established by Ch.J. de la Vallée-Poussin [1]. Cf. Distribution of prime numbers.

References

[1] Ch.J. de la Vallée-Poussin, "Recherches analytiques sur la théorie des nombers premiers" Ann. Soc. Sci. Bruxelles , 20 (1899) pp. 183–256
[2] Ch.J. de la Vallée-Poussin, "Sur la fonction $\zeta(s)$ de Riemann et la nombre des nombres premiers inférieurs à une limite donnée" Mem. Couronnes Acad. Sci. Belg. , 59 : 1 (1899–1900)
[3] K. Prachar, "Primzahlverteilung" , Springer (1957)

S.M. Vorazhin

The de la Vallée-Poussin alternation theorem: If a sequence of points $ \{ x _ {i} \} $, $ i = 0 \dots n + 1 $, in a closed set $ Q \in [a, b] $ forms an alternation, then for the best approximation of a function $ f $ by polynomials of the form

$$ P _ {n} (x) = \ \sum _ {k = 0 } ^ { n } c _ {k} s _ {k} (x), $$

the estimate

$$ E _ {n} (f ) = \ \inf _ {c _ {k} } \ \sup _ {x \in Q } \ \left | f (x) - \sum _ {k = 0 } ^ { n } c _ {k} s _ {k} (x) \right | \geq $$

$$ \geq \ \mathop{\rm min} _ {0 \leq i \leq n + 1 } \ | f (x _ {i} ) - P _ {n} (x _ {i} ) | $$

is valid, where $ {\{ s _ {k} (x) \} } _ {0} ^ {n} $ is a Chebyshev system. Established by Ch.J. de la Vallée-Poussin [1].

According to the Chebyshev theorem, equality holds if and only if $ P _ {n} (x) $ is the polynomial of best approximation. Analogues of this theorem exist for arbitrary Banach spaces [2]. The theorem is employed in numerical methods for constructing polynomials of best approximation.

References

[1] Ch.J. de la Vallée-Poussin, "Sur les polynômes d'approximation et la répresentation approchée d'un angle" Bull. Acad. Belg. , 12 (1910) pp. 808–845
[2] A.L. Garkavi, "The theory of approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian)

Yu.N. Subbotin

Comments

An account of the life and work of de la Vallée-Poussin can be found in, e.g., [a1].

A sequence of points $ x _ {i} $, $ a \leq x _ {1} < \dots < x _ {m} \leq b $, is called an alternation for a continuous function $ g $ on $ [ a , b ] $ if $ g ( x _ {i} ) = ( - 1 ) ^ {i} \| g \| $ where $ \| g \| = \max _ {x \in [ a , b ] } | g (x) | $.

References

[a1] J. Favard, "Hommage à Charles de la Vallée Poussin (1866–1962)" P.L. Butzer (ed.) J. Korevaar (ed.) , On approximation theory , Birkhäuser (1964) pp. 1–3
[a2] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
How to Cite This Entry:
De la Vallée-Poussin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_theorem&oldid=14929
This article was adapted from an original article by S.M. Vorazhin, Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article