Unconditional Convergence of Functional Series in Problems of Probability Theory

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

Quantitative affine approximation for UMD targets, Discrete Analysis 2016:6, 48 pp. Let $Y$ be a Banach space. A _martingale difference sequence_ in $Y$ is a sequence of $Y$-valued random variables such that $\mathbb{E}[d_i|d_1,\dots,d_{i-1}]=0$ for every $i$ (and $\mathbb{E}(d_1)=0$). Given $1<p<\infty$, we say that $Y$ satisfies the UMD$_p$ property if there is a constant $C_p$ such that $$\mathbb{E}\|\sum_{i=1}^n\epsilon_id_i\|^p\leq C_p^p\mathbb{E}\|\sum_{i=1}^nd_i\|^p$$ for every martingale difference sequence $d_1,\dots,d_n$ and every choice of signs $\epsilon_1,\dots,\epsilon_n\in\{-1,1_{}\}$. An _unconditional martingale difference_ space, or UMD space for short, is a Banach space $Y$ that satisfies the UMD$_p$ property for some $1<p<\infty$, which can be shown to imply that it satisfies the UMD$_p$ property for all such $p$. It is easy to show that when $Y$ is a Hilbert space, it has the UMD$_p$ property with a constant of 1. In the context of stochastic analysis, UMD spaces can be thought of as spaces that have many of the good properties of Hilbert spaces. The question considered in this paper concerns a kind of differentiability property of Lipschitz functions. Let $X$ be an $n$-dimensional normed space with unit ball $B_X$ and let $Y$ be a UMD Banach space. Given a 1-Lipschitz function $f:B_X\to Y$ and $\epsilon>0$, the aim is to find a sub-ball $B'\subset B$ of radius $\rho$, with $\rho$ as large as possible, and an affine function $\Lambda:B'\to Y$, such that $\|f(x)-\Lambda(x)\|\leq\epsilon\rho$ for every $x\in B'$. Importantly, the lower bound on $\rho$ must be independent of the function $f$. The paper obtains a lower bound of $\exp(-(1/\epsilon)^{cn})$ for a constant $c$ that depends on $Y$ only. This improves on the previously best known bound even when $Y$ is a Hilbert space. In the other direction, the best known upper bound (on how large $\rho$ is in the worst case) is singly exponential in $n$, so there is still a large and very interesting gap. There are several other attractive open problems in the paper. The proof of the main result of the paper is not easy, but it contains many new ideas in the geometry of Banach spaces and in harmonic analysis, some of which are interesting results in their own right.

The purpose of this work is to study various geometric properties of Banach spaces, with particular emphasis on their stability with respect to the formation of (infinite) absolute sums and Köthe-Bochner spaces of vector- valued functions. The first three chapters are devoted to the class of acs (``alternatively convex or smooth'') Banach spaces and their relatives, the luacs and uacs spaces. These notions were introduced by Kadets et al. in 2000 (in connection with the so called anti-Daugavet property) and form common generalisations of the usual convexity and smoothness properties of Banach spaces. In the first chapter, these notions (as well as some other related properties introduced by the author) are studied from a general point of view. For example, some equivalent characterisations are established and results on duality and quotient spaces are discussed. In the second chapter, various results on (mainly infinite) absolute sums of acs spaces and their relatives are proved, while in the third chapter, acs properties of Köthe-Bochner spaces of vector-valued functions are discussed. The fourth chapter is devoted to the so called Opial property and its variants, as well as some other geometric notions (WORTH property and Garcia-Falset coefficient) of Banach spaces, which are all related to the fixed point property for nonexpansive mappings. Some stability results for these properties in certain infinite absolute sums are established. Furthemore, some properties analogous to the Opial property in Lebesgue-Bochner spaces and Ces`aro spaces of vector-valued functions are discussed. The fifth chapter concerns Banach spaces with the so called ``ball generated property'' (BGP). The stability of this property with respect to certain sums of two spaces is proved. The last chapter deals with generalised lush (GL) spaces, which were introduced by Huang et al. in 2013 in connection with the Mazur-Ulam extension problem. Some results on GL-spaces are established. In particular, every M-ideal in a GL-space is again a GL-space.

We study in this paper the finite Jung constant, its interplay with Kottman’s constant and its meaning regarding the geometry of Banach spaces.

We consider two Pythagorean modulus introduced by Gao (2005, 2006) recently. The exact values concerning these modulus for some classical Banach spaces are determined. Some applications in geometry of Banach spaces are also obtained.

Operator-valued Fourier multiplier theorems on Lp( X) and geometry of Banach spaces

&lt;p&gt;&lt;b&gt;We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.&lt;/b&gt;&lt;/p&gt; &lt;p&gt;We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.&lt;/p&gt; &lt;p&gt;However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.&lt;/p&gt; &lt;p&gt;We also investigate the effectivisations of certain classical theorems in Banachspaces.&lt;/p&gt;

The main result is that in every complete locally-bounded linear topological space there exist series which are unconditionally yet not absolutely convergent. Relations between absolute, unconditional and metric convergence of series are studied.

Let $T$ be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series $\sum_{k=0}^\infty \beta_k T^k x$ when $\{\beta_k\}$ is a Kaluza sequence with divergent sum such that $\beta_k \to 0$ and $\sum

A new, nonclassical convergence acceleration concept, called \(\mu \)-acceleration of convergence (where \(\mu \) is a positive monotonically increasing sequence), is introduced and compared with the classical convergence acceleration concept. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Also, regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. As an application, the results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

General information Conditionally convergent series Unconditionally convergent series Some results in the general theory of Banach spaces $M$-cotype and the Orlicz theorem The Steinitz theorem and $B$-convexity.

On suitable Banach spaces, the existence of a certain functional calculus for an operator is equivalent to that operator having a certain integral repreaentation with respect to some family of projections. In this paper we consider the problem of identifying the precise classes of spaces on which these equivalences hold, for several of the most important functional calculi.

This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is super-exponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on N qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces. One relatively non-standard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept. PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68

In this paper the properties of convex set and its inscribed polygons in a two dimensional Euclidean space axe studied. Consequently we applied our results in solving a problem in geometry of Banach space.