Displaying similar documents to “On Hilbert's 17th Problem and Real Nullstellensatz for Global Analytic Functions.”

On the finiteness of Pythagoras numbers of real meromorphic functions

Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz (2010)

Bulletin de la Société Mathématique de France

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We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field...

Ordered analytic Hilbert spaces over the unit disk

Shengzhao Hou, Shuyun Wei (2008)

Studia Mathematica

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Let f, g be in the analytic function ring Hol(𝔻) over the unit disk 𝔻. We say that f ⪯ g if there exist M > 0 and 0 < r < 1 such that |f(z)| ≤ M|g(z)| whenever r < |z| < 1. Let X be a Hilbert space contained in Hol(𝔻). Then X is called an ordered Hilbert space if f ⪯ g and g ∈ X imply f ∈ X. In this note, we mainly study the connection between an ordered analytic Hilbert space and its reproducing kernel. We also consider when an invariant subspace of the whole space...