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The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the polynomials to be zero. The form of the result had been conjectured by Sergei Bernstein before it was proved. The theorem, in a special case, states that a necessary and sufficient condition for the monomials There are also versions for the Lp spaces.

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rdfs:label	Satz von Müntz-Szász (de) Théorème de Müntz (fr) Müntz–Szász theorem (en) Теорема Мюнтца — Саса (ru)
rdfs:comment	Der Satz von Müntz-Szász (englisch Müntz-Szász theorem) ist einer der Approximationssätze des mathematischen Gebiets der Analysis. Er geht auf Arbeiten der beiden Mathematiker Herman (Chaim) Müntz und Otto Szász aus den Jahren 1914 bzw. 1916 zurück. Der Satz behandelt, anschließend an den klassischen Approximationssatz von Weierstraß, die Frage der Bedingungen, unter denen die stetigen komplexwertigen Funktionen auf dem abgeschlossenen Einheitsintervall durch Linearkombinationen geeigneter Potenzfunktionen gleichmäßig approximiert werden können. (de) Теорема Мюнтца — Саса — утверждение о достаточном условии равномерной аппроксимации произвольной непрерывной функциистепенными полиномами и достаточном условии её невозможности. Была доказана Мюнтцем в 1914 г. и Сасом в 1916 г. Играет важную роль в функциональном анализе. (ru) The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the polynomials to be zero. The form of the result had been conjectured by Sergei Bernstein before it was proved. The theorem, in a special case, states that a necessary and sufficient condition for the monomials There are also versions for the Lp spaces. (en) Le théorème de Müntz-Szász est un résultat fondamental de la théorie de l'approximation, conjecturé en 1912 par Sergeï Bernstein et démontré en 1914 par (en). En 1916, Otto Szász l'a étendu à des exposants complexes et en a fourni une preuve plus simple. Pour I un segment quelconque de ℝ, le théorème de Weierstrass assure que toute fonction continue de I dans ℂ est limite uniforme d'une suite de polynômes. (fr)
dcterms:subject	Theorems in approximation theory Functional analysis
Wikipage page ID	3204608 (xsd:integer)
Wikipage revision ID	981955020 (xsd:integer)
Link from a Wikipage to another Wikipage	Monomial Approximation theory Theorems in approximation theory Uniform norm Complex number Constant function Continuous function Lp space Banach space Divergent series Otto Szász Closed interval Functional analysis Large set (combinatorics) Herman Müntz Real part Sergei Bernstein Dense subset Strictly increasing Weierstrass theorem on polynomial approximation
Link from a Wikipage to an external page	http://www.digizeitschriften.de/dms/resolveppn/%3FPPN=GDZPPN002266563 http://quod.lib.umich.edu/u/umhistmath/aca0698.0001.001/316%3Fview=image&size=125 https://zenodo.org/record/1515344
sameAs	Müntz–Szász theorem Müntz–Szász theorem Müntz–Szász theorem Müntz–Szász theorem Müntz–Szász theorem Müntz–Szász theorem Müntz–Szász theorem
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Cite_journal
has abstract	Der Satz von Müntz-Szász (englisch Müntz-Szász theorem) ist einer der Approximationssätze des mathematischen Gebiets der Analysis. Er geht auf Arbeiten der beiden Mathematiker Herman (Chaim) Müntz und Otto Szász aus den Jahren 1914 bzw. 1916 zurück. Der Satz behandelt, anschließend an den klassischen Approximationssatz von Weierstraß, die Frage der Bedingungen, unter denen die stetigen komplexwertigen Funktionen auf dem abgeschlossenen Einheitsintervall durch Linearkombinationen geeigneter Potenzfunktionen gleichmäßig approximiert werden können. (de) The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the polynomials to be zero. The form of the result had been conjectured by Sergei Bernstein before it was proved. The theorem, in a special case, states that a necessary and sufficient condition for the monomials to span a dense subset of the Banach space C[a,b] of all continuous functions with complex number values on the closed interval [a,b] with a > 0, with the uniform norm, is that the sum of the reciprocals, taken over S, should diverge, i.e. S is a large set. For an interval [0, b], the constant functions are necessary: assuming therefore that 0 is in S, the condition on the other exponents is as before. More generally, one can take exponents from any strictly increasing sequence of positive real numbers, and the same result holds. Szász showed that for complex number exponents, the same condition applied to the sequence of real parts. There are also versions for the Lp spaces. (en) Le théorème de Müntz-Szász est un résultat fondamental de la théorie de l'approximation, conjecturé en 1912 par Sergeï Bernstein et démontré en 1914 par (en). En 1916, Otto Szász l'a étendu à des exposants complexes et en a fourni une preuve plus simple. Pour I un segment quelconque de ℝ, le théorème de Weierstrass assure que toute fonction continue de I dans ℂ est limite uniforme d'une suite de polynômes. Le théorème de Müntz-Szász est une généralisation du théorème de Weierstrass, dans le cas où le segment I est positif, avec un ensemble d'« exposants de monômes » différent de celui des entiers naturels, mais satisfaisant une condition analogue à celle de la divergence de la série harmonique. (fr) Теорема Мюнтца — Саса — утверждение о достаточном условии равномерной аппроксимации произвольной непрерывной функциистепенными полиномами и достаточном условии её невозможности. Была доказана Мюнтцем в 1914 г. и Сасом в 1916 г. Играет важную роль в функциональном анализе. (ru)
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is Link from a Wikipage to another Wikipage of	Muntz List of numerical analysis topics Muntz-Szasz theorem Ottó Szász Stone–Weierstrass theorem Szász William A. Barnett Large set (combinatorics) List of theorems Muntz-Szasz Theorem Muntz–Szasz theorem Muentz approximation theorem Müntz-Szász theorem Muntz approximation theorem Muntz theorem Müntz's theorem Müntz-Szasz theorem Müntz approximation theorem
is Wikipage redirect of	Muntz-Szasz theorem Muntz-Szasz Theorem Muntz–Szasz theorem Muentz approximation theorem Müntz-Szász theorem Muntz approximation theorem Muntz theorem Müntz's theorem Müntz-Szasz theorem Müntz approximation theorem
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