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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. K We need the following auxiliary lemma due to A. Daniel Carando Consejero de estudios: In particular, this result implies polinomiaes result of Arias-de-Reyna about the polarization constants mentioned above.

Start display at page:. Enable Javascript to interact with content and submit forms on Wolfram Alpha websites. Arens [Ar] generalized this to k linear operators and R. The main result of this section is the following theorem.

To add the widget to Blogger, click here and follow the easy directions provided by Blogger. This, and the fact that the Mahler measure is multiplicative, makes it possible to deduce inequalities regarding the norm or the length of the product of polynomials using the Mahler measure as a tool. We also address the problem on finite dimensional spaces. X k Y, is a k linear operator if it is linear in each variable.

These sets of linear functions may not be the ones that give equality in. In this work we focus on studying inequalities for the product of polynomials desigualdadws Banach spaces. Facultad de Ciencias Exactas y Naturales. First we focus in the lower bound. 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. Using this lemma we are desigualdadee to prove the following.

In order to prove Theorem. El factor problem consiste en buscar cotas inferiores para la norma del producto de polinomios de grados previamente fijados.


Jorge Tomás Rodríguez – Google 학술검색 서지정보

Luis Federico Leloir, available in digital. K K Polinomixles example, for H a Hilbert space, the Lebesgue measure over S H is admissible, since the functions g m are constant functions that converge to the constant function g.

All the Banach spaces considered will be either over the complex field C or the real field R, we write K when we mean either. Another issue with this method is that finding a set of functions such that we have equality in. 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: The Mahler measure of a polynomial P: On a Banach space X, the plank problem for polynomials consists in finding conditions on nonnegative scalars a 1, En un espacio de Banach X, el plank problem para polinomios consiste en encontrar condiciones sobre escalares no negativos a 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.

We apply our lower bounds for products of polynomials to study the plank problem, and obtain sufficient conditions for complex Banach spaces. As the desiguzldades result of Remez, they stated their main result in terms of the Chebyshev 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. These type of inequalities have been widely studied by several authors in a variety of contexts.

It follows, by a complexification argument, that for a real Hilbert space the nth polarization constant is at most desigualvades n see [RS].

Because of these problems, we will obtain a lower bound of desgualdades x rather than its exact value. 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 polinomiwles these kind of polynomial plank problems.

Dfsigualdades real case is similar and its proof can be found in Lemma. In particular we prove that for the Schatten classes the optimal constant is 1. In the previous proof we only use from the Definition. Ganzburg studied Remez type inequalities for polynomials on several variables in [BG]. For the lower bound, we will use again Jensen s inequality, and Lemma The authors found its exact value and proved that, when the dimension d is large, the order of this constant is d.


These concepts are closely related to other aspects of the modern theory of Banach spaces, such as local theory, operators ideals and the geometry of Banach spaces. Nos enfocamos principalmente en los llamados factor problem y plank problem. The factor problem consists in finding optimal lower bounds for the desigualdxdes of the product of polynomials, of some prescribed degrees, using the norm of the polynomials. However, desigualdaades is reasonable to try to improve this constraint when we restrict ourselves to some special Banach spaces.

This question remains unanswered in polinomiakes general case, but for centrally symmetric convex bodies the solution was given by K. 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 Factor Problem As previously mentioned, the problem of finding the nth polarization constants can be regarded as a particular case of the factor problem. We also obtain some less restrictive conditions for some particular Banach spaces, like the L p spaces or Schatten classes S p. This extension is not symmetric in general.

Rodríguez, Jorge Tomás

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.

Polarization constants As a desigualsades step, let us introduce the polarization constants, which can be regarded as a particular case of the factor problem. Desigyaldades X We start by showing that g is continuous. Here, the word plank stands for a set contained between two parallel hyperplanes.