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.
Müntz, Ch. H. (1914). "Über den Approximationssatz von Weierstrass". H. A. Schwarz's Festschrift. Berlin. pp. 303–312.{{cite book}}: CS1 maint: location missing publisher (link) Scanned at University of Michigan
Szász, O. (1916). "Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen". Math. Ann. 77 (4): 482–496. doi:10.1007/BF01456964. S2CID 123893394. Scanned at digizeitschriften.de
Shen, Jie; Wang, Yingwei (2016). "Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems". SIAM Journal on Scientific Computing. 38 (4): A2357–A2381. Bibcode:2016SJSC...38A2357S. doi:10.1137/15M1052391.
January 01, 1970
müntz, szász, theorem, basic, result, approximation, theory, proved, herman, müntz, 1914, otto, szász, 1884, 1952, 1916, roughly, speaking, theorem, shows, what, extent, weierstrass, theorem, polynomial, approximation, have, holes, into, restricting, certain, . The Muntz Szasz theorem is a basic result of approximation theory proved by Herman Muntz in 1914 and Otto Szasz 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 x n n S N displaystyle x n quad n in S subset mathbb N 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 gt 0 with the uniform norm is that the sum n S 1 n displaystyle sum n in S frac 1 n 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 Szasz showed that for complex number exponents the same condition applied to the sequence of real parts There are also versions for the Lp spaces See also editErdos conjecture on arithmetic progressionsReferences editMuntz Ch H 1914 Uber den Approximationssatz von Weierstrass H A Schwarz s Festschrift Berlin pp 303 312 a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link Scanned at University of Michigan Szasz O 1916 Uber die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen Math Ann 77 4 482 496 doi 10 1007 BF01456964 S2CID 123893394 Scanned at digizeitschriften de Shen Jie Wang Yingwei 2016 Muntz Galerkin methods and applications to mixed Dirichlet Neumann boundary value problems SIAM Journal on Scientific Computing 38 4 A2357 A2381 Bibcode 2016SJSC 38A2357S doi 10 1137 15M1052391 Retrieved from https en wikipedia org w index php title Muntz Szasz theorem amp oldid 1180537782, wikipedia, wiki, book, books, library,
