Geraldo Botelho, Daniel Pellegrino, Pilar Rueda
The aim of this paper is to prove a stronger version of a conjecture on the existence of non-dominated scalar-valued m-homogeneous polynomials (m>=3) on arbitrary infinite dimensional Banach spaces.
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In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However, this alternative scheme is shown not to be satisfactory until the equivalence is proved.
We relate the injectivity of the canonical map from $C(B_{E'})$ to $L_p(μ)$, where $μ$ is a regular Borel probability measure on the closed unit ball $B_{E'}$ of the dual $E'$ of a Banach space $E$ endowed with the weak* topology, to the existence of injective $p$-summing linear operators/$p$-dominated homogeneous polynomials defined on $E$ having $μ$ as a Pietsch measure. As an application we fill the gap in the proofs of some results of concerning Pietsch-type factorization of dominated polynomials.
The main goal of this note is to establish a connection between the cotype of the Banach space X and the parameters r for which every 2-homogeneous polynomial on X is r-dominated. Let cotX be the infimum of the cotypes assumed by X and (cotX)* be its conjugate. The main result of this note asserts that if cotX > 2, then for every 1<= r < (cotX)* there exists a non-r-dominated 2-homogeneous polynomial on X.
Let T and C be two Hilbert space operators. We prove that if T is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by C, then T is similar to an operator which is completely polynomially dominated by the direct sum of C and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pi…
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution of a general class of high order heat-type equations.
It is known that many constructions arising in the classical Gaussian infinite dimensional analysis can be extended to the case of more general measures. One such extension can be obtained through biorthogonal systems of Appell polynomials and generalized functions. In this paper, we consider linear continuous operators from a nuclear Frechet space of test functions to itself in this more general setting. We construct an isometric integral transform (biorthogonal CS-transform) of those operators into the space of germs of holomorphic functions on a locally convex infinite dimensional nuclear s…
Zero sets of homogeneous polynomials containing infinite dimensional spaces
本文研究了无限维线性空间上齐次多项式零点集的性质，证明了若多项式在有限维子空间为零，则存在包含该子空间的无限维子空间使其为零，并推广了复情形的经典结果。
In this paper, we establish two different characterizations of Walrasian expectations allocations by the veto power of the grand coalition in an asymmetric information economy having finite numbers of agents and states of nature and whose commodity space is a Banach lattice. The first one deals with Aubin non-dominated allocations, and the other claims that an allocation is a Walrasian expectations allocation if and only if it is not privately dominated by the grand coalition, by considering perturbations of the original initial endowments in precise directions.
In this note we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach multi-ideal which is a holomorphy type (in the sense of L.Nachbin) and also coherent and compatible (in the sense of D. Carando, V. Dimant and S. Muro) with the notion of almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho.
The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for applications in Harmonic Analysis and Number Theory. Up to now, the best known estimates for its constants are dominated by $κ\left(1+\varepsilon\right) ^{m}$, where $\varepsilon>0$ is arbitrary and $κ>0$ depends on the choice of $\varepsilon$. For the special cases in which the number of variables in each monomial is bounded by some fixed number $M$, it has b…
The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence…
Zeros of orthogonal polynomials and some matrix inequalities
本研究利用正交多项式的矩阵方法、有界点赋值概念与Wirtinger型不等式，证明了多类向量测度下Sobolev正交多项式的零点有界性，突破了序列控制的前提限制。
Linear equations and multiplicative polynomial equations in infinitely many variables
本文研究无穷变量多项式方程组，证明若每个有限子集有解或近似解，则整个无穷方程组有解。
We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincaré-type polynomials. It turns out that the fulfillment of most of types of decoupling inequalities may depend on the geometry of Banach space.
The purpose of this paper is to study some new concrete approximation processes for continuous vector-valued mappings defined on the infinite dimensional cube or on a subset of a real Hilbert space. In both cases these operators are modelled on classical Bernstein polynomials and represent a possible extension to an infinite dimensional setting. The same idea is generalized to obtain from a given approximation process for function defined on a real interval a new approximation process for vector-valued mappings defined on subsets of a real Hilbert space.
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinit…
Orthogonal Intertwiners for Infinite Particle Systems in The Continuum
本文证明无限维正交多项式交织无限粒子系统（粘性布朗运动及相关布朗运动）与其n粒子演化，并利用新显式公式获得新的可逆测度。
Sard properties for polynomial maps in infinite dimension
本文研究了无穷维多项式映射的Sard性质，提出了Sard定理在该设定下的精确量化条件，并应用于亚黎曼几何中的Sard猜想证明。
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