Müntz–Szász theorem

Müntz–Szász theorem The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) dentro 1916. A grosso modo, 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.

O teorema, in a special case, states that a necessary and sufficient condition for the monomials {estilo de exibição x^{n},quad nin Ssubset mathbb {N} } to span a dense subset of the Banach space C[uma,b] of all continuous functions with complex number values on the closed interval [uma,b] with a > 0, with the uniform norm, is that the sum {soma de estilo de exibição _{nin S}{fratura {1}{n}} } of the reciprocals, taken over S, should diverge, ou seja. 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.

De forma geral, 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.

References Müntz, CH. H. (1914). "Über den Approximationssatz von Weierstrass". H. UMA. Schwarz's Festschrift. Berlim. pp. 303–312. Scanned at University of Michigan Szász, O. (1916). "Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen". Matemática. Ana. 77: 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. doi:10.1137/15M1052391. Categorias: Functional analysisTheorems in approximation theory