Holomorphic Functions on Nuclear Sequence Spaces

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition t

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

In this article we discuss the relationship between domains of existence domains of holomorphy, holomorphically convex domains, pseudo convex domains, in the context of locally convex topological vector spaces. By using the method of Hirschowitz for Π n=1 ∞ C and the method used for Banach spaces with a basis we prove generalisations of the Cartan-Thullen-Oka-Norguet-Bremmerman theorem for finitely polynomially convex domains in a variety of locally convex spaces which include the following:1) N-projective limits of Frechet spaces with a basis;2) countable direct sums of Frechet spaces with a basis;3) nuclear spaces.

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

Over the past years, a significant interest has developed in the study of holomorphic functions defined on a domain in an infinite-dimensional Banach space and of their constituents (via Taylor expansions), the homogeneous polynomials. Many of the questions that have been studied have arisen from considerations of infinitedimensional topology and from standard function algebra questions [CCG]. Recently, there has been an interest in connecting the well-developed theory of the geometry of Banach spaces with the function theory questions that have been studied classically, and some progress has been made in this direction [ACG; D; F; CCG; CGJ]. In addition, connections between properties of polynomials and geometry of the unit ball has been of interest (see [GJL] for a survey of this topic). The present work is an attempt to study some of the properties of bounded analytic functions on the unit ball of an infinite-dimensional Banach space. In particular, we are interested in understanding something of boundary behavior; we combine techniques from the several fields to investigate it, especially with regard to the interplay with convexity and smoothness. Many of the results here apply to the classical “nice” reflexive spaces, such as lp and Lp (1 < p <∞). It is almost certain that there is much more to be learned even about the Hilbert space case. We consider the boundary behavior of H∞ functions on B, the open unit ball of an infinite-dimensional complex Banach space that has the geometric properties of uniform convexity, uniform smoothness, or both. By uniform smoothness, we mean uniform (real) Frechet differentiability of the norm, with the space considered as a real Banach space. Uniform convexity will mean that the dual is uniformly smooth; since spaces with either property are reflexive, this definition is complete. To be specific however, we state the following (after [LT]).

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.

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 investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.

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 structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

We study the unconditional convergence of series in Banach spaces. We consider series of special type (Hadamard series), obtain the condition of their unconditional convergence, and discuss some of their applications. Further, we examine the almost sure unconditional convergence of random series in Banach spaces and, in the case of Gaussian series, we establish the relationship between the almost sure unconditional convergence and the geometry of Banach spaces. We also consider some probabilistic problems related to the convergence of series.

Probabilistic methods in the geometry of Banach spaces