# De la Vallée-Poussin theorem

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.

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#### 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

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 $\| g \| = \max _ {x \in [ a , b ] } | g (x) |$.