DESIGUALDADES POLINOMIALES PDF

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Luis Federico Leloir, disponible en digital. This document is part of the doctoral theses collection of the Central Library Dr. Luis Polinomiakes Leloir, available in digital. It should be used accompanied by the corresponding citation acknowledging the source. Desigualdades polinomiales en espacios de Banach. Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires. Daniel Carando Consejero de estudios: Daniel Carando Lugar de trabajo: Nos enfocamos principalmente en los llamados factor problem y plank dezigualdades.

El factor problem consiste en buscar cotas inferiores para la norma del producto de polinomios de grados previamente fijados. Estudiamos este problema en diferentes contextos.

Consideramos el producto de funciones lineales i. En un espacio de Banach X, el plank problem para polinomios consiste en encontrar condiciones sobre escalares no negativos a 1, Aplicamos las cotas inferiores obtenidas para el producto de polinomios al estudio de este problema y obtenemos condiciones suficientes para espacios de Banach complejos. We focus mainly on the so called factor problem and plank problem.

The factor problem is the problem of finding lower bounds for the norm of the product of polynomials of some prescribed degrees. We study this problem in different contexts. We consider the product of linear functions i. We also study this problem on different spaces: Insomecase, likeinthel p spacesandtheschattenclassess p, we obtain optimal lower bounds, while for other spaces we only give some estimates of the optimal lower bounds.

On a Banach space X, the plank problem for polynomials consists in finding conditions on nonnegative scalars a 1, We apply our lower bounds for products of polynomials to study the plank problem, and obtain sufficient conditions polinomkales complex Banach spaces.

We also obtain desigualdaades less restrictive conditions for some particular Banach spaces, like the L p spaces or Schatten classes S p. Mis padres, por su constante apoyo durante mis estudios. Dedigualdades hermanos, quienes siempre me dieron una mano cuando la necesite. Lucia, polinomixles estar siempre.

Por su interminable paciencia y estar siempre presentes para aconsejarme y asistirme. Por ayudarme a reforzar mis cualidades y atenuar algunos de mis defectos. One of the most important asset polynomials have is the large amount of functions they can approximate and that they are much easier to handle than these functions.

In particular, in Functional Analysis, the study of multilinear forms and polynomials has been growing in the last decades. These concepts are closely related to other aspects of the modern theory of Banach spaces, polinomialez as local theory, operators ideals and the geometry of Banach spaces. ;olinomiales

Rodríguez, Jorge Tomás

In this work we lolinomiales on studying inequalities for the product of polynomials on Banach spaces. We study what is sometimes called the factor problem and its applications to a geometrical problem called the plank problem.

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Although these problems are related, they are of an independent interest and will be treated accordingly. Below, we provide a brief introduction to the factor problem and the plank problem, as well as some of the results already known regarding these problems. Polarization constants As a first desugualdades, let us introduce the desigualcades constants, which can be regarded as a particular case of the factor problem.

Turett proved a sort of reverse inequality: Later, as a corollary of Theorem 3 from [BST], it desigualdases obtained that the optimal K n with such property, for complex Banach spaces, is n n. It follows, by a complexification argument, that for a real Hilbert space the nth polarization constant is at most n n see [RS]. It is conjectured, for example, that the result of Arias-de-Reyna holds for real Desiguladades spaces.

That is, this constant is determined by the behaviour of the nth polarization constants when n is large. The authors found the polinomialez value of c r d and proved that its order is d. This result was later extended to complex Hilbert desigualldades by A. We devote Chapter to analyze these constants. 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.

The Factor Problem As previously mentioned, the problem of finding the nth polarization constants can be regarded as a particular case of the factor problem. The factor problem consists in finding optimal lower bounds for the norm of the product of polynomials, of some dexigualdades degrees, using the norm of the polynomials.

This problem has been studied in several spaces, considering a wide variety of norms. For example, in [B], W. Boyd proved that for any set of polynomials P 1, In Chapter 3 we study the factor problem on Banach spaces, considering the standard uniform norm for polynomials.

X K, of degrees k 1, As pointed out before, a corollary from this result is that c n X n n for any complex Banach space. The authors also showed that this is the best universal constant, since there are polynomials on l 1 for which equality prevails.

However, as we will see later, in some cases it is possible to improve this bound. In Chapter 3 we study the factor problem on several spaces. Exploiting the similarities between the L p spaces and the Schatten classes S p, we are able to transport some results, obtained for the L p spaces, to the Schatten classes. In particular we prove that for the Schatten classes the optimal constant is lolinomiales.

In this chapter we also study the factor problem on ultraproducts of Banach spaces. 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.

Using similar ideas, we show that given a Banach space X such that X has the metric approximation property, then the best constant for X and X is the same. Given a convex body K R d of minimal width 1, if K is covered by n planks with widths a 1, Here, the word plank stands for a set contained between two parallel hyperplanes. The solution to this problem was given by T. The author also presented a similar question to Tarski s Plank Problem.

DESIGUALDADES CON VALOR ABSOLUTO – Casos 2 y 3

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? The relative width of a plank is the width of the plank divided by the width of the convex body in the direction that the plank attains its width.

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This question remains unanswered in the general case, but for centrally symmetric convex bodies the solution was given by K. Ball in [Ba1], where he proved slightly more than the following: However, it is reasonable to try to improve this constraint when we restrict ourselves to some special Banach spaces.

Unfortunately, this stronger result is not necessarily valid if we choose other sets of unit functionals.

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Nonetheless, the situation is quite different if we consider complex Hilbert spaces, as K. Ball showed in [Ba]: In particular, this result implies the result of Arias-de-Reyna about the polarization constants mentioned above.

Kim [KK] studied a plank type problem for polynomials on Banach spaces. 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. We aim to give sufficient conditions such that if a 1, We also address the problem on finite dimensional spaces. Given a Banach space X, we denote its topological dual by X, the norm on the space by X or, when it is clear from context.

All the Banach spaces desigualdaees will be either over the complex field C or the real field R, we write Desigualdadew when we mean either. The later will be used mostly when we have both situations: For multilinear operators between two spaces, we will use T and reserve the letters P and Q for polynomials.

In the particular case when X is a real Hilbert space of dimension d we write B d and S d 1 instead. In order to introduce the notion of polynomials on Banach spaces we start by 7. X k Y, is a k linear operator if it is linear in each variable. The space of k linear continuous operators from X Definition Given k N, a mapping P: Note that for a finite dimensional space K d,this definition agrees with the standard definition of a polynomial on several variables, where a mapping P: Then, for any x 0 X, we have the following formula for the operator T: For details on this classical result, as well as a deeper introduction to polynomials on Banach spaces, we refer the reader to the survey by Richard Aron [Arn] on this topic.

Arens [Ar] generalized this to k linear operators and R. Berner [AB] to homogeneous polynomials, with the method described below. First, define a function z: Similarly, we can define z: This extension is not symmetric in general.

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Actually, it does depend on the oolinomiales we do the composition z 1 z k. But the restriction of AB T to the diagonal does not depend on the order of the composition. For more details on the Aron-Berner extension, as well as extensions of polynomials in general, we refer the reader to the survey by I.