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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 $x \rightarrow \infty$,
$$
\pi(x) \sim \frac{x}{\log x} \ .
$$

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

====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 $\zeta(s)$ 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''

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

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

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 [[#References|[1]]].

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.L. Garkavi,   "The theory of approximation in normed linear spaces"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 75–132  (In Russian)</TD></TR></table>

''Yu.N. Subbotin''

====Comments====
An account of the life and work of de la Vallée-Poussin can be found in, e.g., [[#References|[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====
<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>