Difference between revisions of "De la Vallée-Poussin theorem"
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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: | 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: | ||
$$ | $$ | ||
| Line 19: | Line 21: | ||
''S.M. Vorazhin'' | ''S.M. Vorazhin'' | ||
| − | The de la Vallée-Poussin alternation theorem: If a sequence of points | + | 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 | 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 | + | 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 | + | 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==== | ||
| Line 41: | Line 63: | ||
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 < | + | 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==== | ||
Revision as of 12:34, 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 |
De la Vallée-Poussin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_theorem&oldid=42049