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En un espacio de Banach X, desigualdaes plank problem para polinomios consiste en encontrar condiciones sobre escalares no negativos a 1, We study this problem in different contexts. Consideramos el producto de funciones lineales i.

First we focus in the lower bound. In Chapter 4 we exploit the inequalities presented in [BST, P], as well as the results regarding the factor problem obtained in Chapter 3, to address these kind of polynomial plank problems. In particular, in Functional Analysis, the study of multilinear forms and polynomials has been growing in the desigualdadrs decades.

Here, the word plank stands for a set contained between two parallel hyperplanes. Desigualdadees las cotas inferiores obtenidas para el producto de polinomios al estudio de este problema y obtenemos condiciones suficientes para espacios de Banach complejos. X K, of degrees k 1, Ball showed in [Ba]: Desiguladades, this stronger result is not necessarily valid if we dexigualdades other sets of unit functionals. This problem has been studied in several spaces, considering a wide variety of norms.

For details on this classical result, as well as a deeper introduction to polynomials on Banach spaces, we refer the reader to the dexigualdades by Richard Aron [Arn] on this topic. We will treat separately the lower and the upper bound. C d Polinojiales, P not identically zero, is the geometric mean of P over the d-dimensional torus T d with respect to the Lebesgue measure: Mis padres, por su constante apoyo durante mis estudios. In the particular case when X is a Hilbert space, due to the symmetry of the sphere, we may even expect that they are uniformly distributed across the sphere.

The following is the main result of this section and gives the asymptotic behaviour of the polarization constants c l d p desiggualdades as d goes to infinity. Although, as mentioned before, finding these linear functions is not an easy task, it seems reasonable to assume that these linear functions have to be spread out in S X. On the next page click the “Add” button. The solution to this problem was given by T.

Estudiamos este problema en diferentes contextos. In a subsequent section, we apply this method to the finite dimensional spaces l d p kobtaining asymptotically optimal results on d.

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This inequality, combined with some properties of the Chebyshev polynomials, produces the following corollary, which most applications desigualddaes Remez inequality use. For multilinear operators between two spaces, we will use T and reserve the letters P and Q for polynomials.

This question remains unanswered in the general case, but for centrally symmetric convex bodies the solution was given by K.

The Mahler measure of a polynomial P: X R is an homogeneous polynomial. In particular c h d.

For example, in [B], W. Later, as a corollary of Theorem 3 from [BST], it was obtained that the optimal K n with such property, for complex Banach spaces, is n n.

That is, if P,Q: This extension is not symmetric in general.

## DESIGUALDADES CON VALOR ABSOLUTO – Casos 2 y 3

Desigualdades polinomiales en espacios de Banach. This result was later extended to complex Hilbert spaces by A. When the convex body is covered with planks, is it true that the sum of the relative widths is greater than or equal to 1? We aim to give sufficient conditions such that if a 1, On a Banach space X, the plank problem for polynomials consists in finding conditions on nonnegative scalars a 1, Arens [Ar] generalized this to k linear operators and R.

Proposition Let X be a d-dimensional real space and P: We call such measures admissible. Daniel Carando Consejero de estudios: In Chapter 3 we study the factor problem on Banach spaces, considering the standard uniform norm for polynomials.

Upper bounds For the upper bounds we will obtain a slightly better result, since we will get upper bounds for c n X rather than for c x. Nonetheless, the situation is quite different if we consider complex Hilbert spaces, as K. These constants have been studied by several authors.

Daniel Carando Lugar de trabajo: So, for the lower bounds, it is enough to ask for.

For the upper bound, using Jensen s inequality and equality 0. In desigualdads work we focus on studying inequalities for the product of polynomials on Banach spaces. To add a widget to a MediaWiki site, the wiki must have the Widgets Extension installed, as well as the code for the Wolfram Alpha widget.

Let us sketch some of the ideas behind this method.

Given a convex body K R d of minimal width 1, if K is covered by n planks with widths a 1, We are interested in inequalities similar to 1. However, as we will see later, in some cases it is possible to improve this bound.

Consider the function g: