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.

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