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Nonetheless, the situation is quite different if we consider complex Hilbert spaces, as K. We devote the first section to develop a method to estimate the polarization constant of a finite dimensional space.

This result holds for real Hilbert spaces and n 5 see [PR], Theorembut it is not known if it is true for every natural number n. Start display at page:. Here, the word plank stands for a set contained between two parallel hyperplanes. These constants have been studied by several authors.

### Jorge Tomás Rodríguez – Google Scholar Citations

Luis Federico Leloir, available in digital. The factor problem consists in finding optimal lower bounds for plinomiales norm of the product of polynomials, of some prescribed degrees, using the norm of the polynomials.

In particular, in Functional Analysis, the study of multilinear forms and polinomiakes has been growing in the last decades. Turett proved a sort of reverse inequality: We focus mainly on the so called factor problem and plank problem.

Note that for a finite dimensional space K d,this definition agrees with the standard definition of a desigialdades on several variables, where a mapping P: It is important to remark that this terminology it is not standard an in some works the polarization constant stands for a different constant, see for example [Di] and [LR].

The authors also showed that this is the best universal constant, since there are polynomials on l 1 for which equality prevails. A Banach space X is finite C d C, P not identically zero, is the geometric mean of P over the d-dimensional torus T d with pokinomiales to the Lebesgue measure: Because of these problems, we will desgualdades a lower bound of c x rather than its exact value.

As mentioned before, we want to consider a measure related to the geometry of the sphere S l d p R. Let us sketch some of the ideas behind this method. To include the widget in a wiki page, paste the code below into the page source.

### “Desigualdad polinomial” – Google Search

Given an ultraproduct of Banach spaces X i U we prove that under certain conditions the best constant for this ultraproduct is the limit of the best constants for the spaces X i. However, it is reasonable to try lolinomiales improve this constraint when we restrict ourselves to some special Banach spaces. We appreciate your interest in Wolfram Alpha and will be in touch soon. In particular we prove that for the Schatten classes the optimal constant is 1.

This problem has been studied in several spaces, considering a wide variety of norms. Enable Javascript to interact with content and submit forms on Wolfram Alpha websites. The real case is similar and its proof can be found in Lemma.

In order to introduce the notion of polynomials on Banach spaces polinomiale start by 7. The main result of this section is the following theorem. The following is the main result of this section and gives the asymptotic behaviour of the polarization constants c l d p k as d goes to infinity.

Aplicamos las cotas inferiores obtenidas para el producto de polinomios al estudio de este problema y obtenemos condiciones suficientes para espacios de Banach complejos. The space of k linear continuous operators from X That being said, we also want a measure that can be easily related to the Lebesgue measure of S d 1, given that for Hilbert spaces the polarization constant is known. En este contexto utilizamos los resultados presentados en [BG] por Y. X k Y, is a k linear operator if it is linear in each variable.

The Mahler measure M P of a polynomial P: More details on the Mahler measure can be found in the work of of M. In a subsequent section, we apply this method to the finite dimensional spaces l d p kobtaining asymptotically optimal results on d. In particular c h d. X K, of degrees k 1, Among the works on this topic, in [RT] the authors proved that for each n there is a constant K n such that c n X K n for every Banach space X.

## DESIGUALDADES CON VALOR ABSOLUTO – Casos 2 y 3

Unfortunately, this stronger result is not necessarily valid if we desigaldades other sets of unit functionals. For the upper bound, using Jensen s inequality and equality 0.

Remez type inequalities In this section, we introduce Remez type inequalities for polynomials. E K es un polinomio continuo, con norma igual a la de P, que coincide con P sobre E pensado como subespacio de E. In Chapter 3 we study the factor problem on several spaces. For multilinear operators between two spaces, we will use T and reserve the letters P and Q for polynomials. Given a convex body K R d of minimal width 1, if K is covered by n planks with widths a 1, Using this lemma we are able to prove the following.

Universidad de Buenos Aires. This extension is not symmetric in general. We aim to give sufficient conditions such that if a 1, We also give some estimates on the norm of the product of linear functions on l d Cthus obtaining bounds for the nth polarization constant c n l d C.

It is conjectured, for example, that the result of Arias-de-Reyna holds for real Hilbert spaces. In this chapter we study the nth polarization constants, as well as the polarization constant, of finite dimensional Banach spaces. First we focus in the lower bound.

We study what is sometimes called the factor problem and its applications to a geometrical problem called the plank problem.