Bappaditya Bhowmik, Nilanjan Das
We explore the Bohr inequality involving the Fourier transforms of complex valued integrable and square integrable functions defined on a second countable compact topological group. We also investigate the connection of the Bohr phenomenon with a modulus of convexity of the space of bounded linear operators defined on a complex Hilbert space.
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The purpose of this article is to study Bohr inequalities involving the absolute values of the coefficients of an operator valued function. To be more specific, we establish an operator valued analogue of a classical result regarding the Bohr phenomenon for scalar valued functions with fixed initial coefficient. Apart from that, operator valued versions of other related and well known results are obtained.
In this article we establish Bohr inequalities for operator valued functions, which can be viewed as the analogues of a couple of interesting results from scalar valued settings. Some results of this paper are motivated by the classical flavor of Bohr inequality, while the others are based on a generalized concept of the Bohr radius problem.
Local Banach Space Theoretic Approach to Bohr's Theorem for Vector Valued Holomorphic and Pluriharmonic Functions
本文通过局部巴拿赫空间理论研究向量值全纯和多值调和函数的波尔定理，证明了波尔半径的正性及渐进行为，并建立了相关系数型施瓦茨-皮克引理。
In this article, we study the multi-dimensional Bohr radii of holomorphic functions defined on the Banach sequence spaces with values in the Banach spaces. For the case of finite dimensional Banach spaces, we exhibit the exact asymptotic growth of the Bohr radius. To achieve our goal in the finite case, we use $\ell_{p'}$-summability of certain coefficients of a given polynomial in terms of its uniform norm on $\ell_p^n$. The infinite case is handled using the techniques developed in recent years from the work of Defant, Maestre and Schwarting. We crucially use several properties of the symmet…
Operator valued analogues of multidimensional Bohr's inequality
本文研究了算子值多维波尔不等式的改进与精炼版本，针对复 Hilbert 空间上的有界解析函数，证明了在完全圆域 Q 中的多维算子不等式。
Bohr operator on opertor valued polyanalytic functions on simply connected domains
本文研究了在单位盘上由f子ordinating的算子值子ordinating类S(f)的Bohr算子，推导了子ordinating结果，并获得了自分析映射的von Neumann型不等式，还研究了算子值多分析函数的Bohr不等式，求得了特定形式的Bohr半径。
We study a "$p$-powered" version $K_n^p(F(R))$ of the well-known Bohr radius problem for the family $F(R)$ of holomorphic functions $f: R\to X$ satisfying $\|f\|<\infty$, where $\|.\|$ is a norm in the function space $F(R)$, $R\subset\mathbb{C}^n$ is a complete Reinhardt domain and $X$ is a complex Banach space. For all $p>0$, we describe in full details the asymptotic behaviour of $K_n^p(F(R))$, where $F(R)$ is (a) the Hardy space of $X$-valued holomorphic functions defined in the open unit polydisk $\mathbb{D}^n$, and (b) the space of bounded $X$-valued holomorphic or complex-valued plurihar…
Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials
本研究构建通用框架分析多元解析多项式Banach空间的投影常数等基本常数的渐近最优行为，推导一般Banach序列格的Bohr半径估计，并在有限维Lorentz序列空间中得到宽参数范围的最优结果。
The generalized Bohr radius $R_{p, q}(X), p, q\in[1, \infty)$ for a complex Banach space $X$ was introduced by Blasco in 2010. In this article, we determine the exact value of $R_{p, q}(\mathbb{C})$ for the cases (i) $p, q\in[1, 2]$, (ii) $p\in (2, \infty), q\in [1, 2]$ and (iii) $p, q\in [2, \infty)$. Moreover, we consider an $n$-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal{H})$ for an infinite dimensional complex Hilbert space $\mathcal{H}$, (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to\infty$ for finite dimensional $X$…
Operator-valued multivariable Bohr type inequalities are obtained for: a class of noncommutative holomorphic functions, generalizing the analytic functions on the open unit disc; the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra; a class of noncommutative selfadjoint harmonic functions, generalizing the real-valued harmonic functions on the open unit disc; Cuntz-Toeplitz algebras ; the reduced (resp.full) group C*-algebra of the free group with n generators; a class of analytic functions on the open unit ball of C^n. The classical Bohr inequality is shown to be a…
We study the asymptotic decay of the Fourier spectrum of real functions $f\colon \{-1,1\}^N \rightarrow \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset \{1,\ldots,N\}}\widehat{f}(S) x^S \,,$ where $x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1, 1\}^{N} $, the Boolean radius of $\mathcal{F}$ is defined to be the largest $ρ\geq 0$ such that $\sum_{S}{|\widehat{f}(S)| ρ^{|S|}} \leq \|f\|_{\infty}$ for every $f \in \…
The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr approximation theorem for almost periodic holomorphic functions on a strip to the case of Banach-valued almost periodic holomorphic functions.
Bohr and Rogosinski inequalities for operator valued holomorphic functions
本文研究了复Banach空间中算子值全纯函数的Bohr半径和p-均匀C-凸性，证明了p-均匀C-凸性等价于Bohr半径正数，并探讨了Lebesgue空间中的相关性质。
The Bohr-Bohnenblust-Hille Theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series $\sum_n a_n n^{-s}$ converges uniformly but not absolutely is less than or equal to 1/2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space $\mathcal{H}_\infty$ equals 1/2. By a surprising fact of Bayart the same result holds true if $\mathcal{H}_\infty$ is replaced by any Hardy space $\mathcal{H}_p$, $1 \le p < \infty$, of Dirichlet series. For Dirichlet series with coefficients i…
Bohr radius for Banach spaces on simply connected domains
研究Banach空间和Hilbert空间上单连通域的Bohr半径，建立算子值Césàro算子和Bernardi算子的Bohr不等式。
Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series $\sum a_n n^{-s}$, and more recently even on general Dirichlet series $\sum a_n e^{-λ_n s}$. This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a …
For $\Cal A\subset L^1_{loc}(\Bbb J,X)$ let $\Cal M\Cal A$ consist of all $f\in L^1_{loc}$ with $ M_h f (\cdot):=\frac {1}{h}\int_{0}^{h}f(\cdot +s)\,ds \in \Cal A$ for all $h>0$. Here $X$ is a Banach space, $\Bbb J= (α,\infty), [α,\infty)$ or $\Bbb R$. Usually $\Cal A\subset\Cal M\Cal A\subset \Cal M^2\Cal A\subset ...$. The map $ \Cal A \to \Cal {D}'_{\Cal A}$ is iteration complete, that is $ \Cal {D}'_{\Cal {D}'_{\Cal A}}= \Cal {D}'_{\Cal A}$. Under suitable assumptions $\widetilde {\Cal M}^n \Cal {A}= \Cal A + \{T^{(n)} : T \in \Cal A\}$, and similarly for $\Cal {M}^n \Cal A$. Almost perio…
For $α\in \mathbb{R}$, let $\mathscr{D}_α$ denote the scale of Hilbert spaces consisting of Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$ that satisfy $\sum_{n=1}^\infty |a_n|^2/[d(n)]^α< \infty$. The Gordon--Hedenmalm Theorem on composition operators for $\mathscr{H}^2=\mathscr{D}_0$ is extended to the Bergman case $α>0$. These composition operators are generated by functions of the form $Φ(s) = c_0 s + φ(s)$, where $c_0$ is a nonnegative integer and $φ(s)$ is a Dirichlet series with certain convergence and mapping properties. For the operators with $c_0=0$ a new phenomenon is discov…
Set-Valued Fractal Approximation for Countable Data Sets
本文利用分形几何方法构造可数数据集上的分形函数，并用于逼近集值映射，同时证明其图像上不变Borel测度的存在唯一性并给出维数有效界。
We consider various notions of holomorphic extendability of complex valued functions defined on subsets of $\mathbf C^n$, including one-sided extendability. We show that in the relevant function spaces, these phenomena of holomorphic extendability are rare in the topological sense, generalizing several results of the article "One sided extendability and $p$-continuous analytic capacities" by E. Bolkas, V. Nestoridis, C. Panagiotis and M. Papadimitrakis, in dimensions $n\ge 2$.
On products and partial isometry of Toeplitz operators with operator-valued symbols
本文刻画了算子值符号Toeplitz算子乘积仍为Toeplitz算子的条件，并证明了部分等距Toeplitz算子可分解为两个算子值内函数对应的乘子及其伴随的乘积，推广了标量值Hardy空间上的经典结果。
We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is…
A relaxation to the post-composition approach to metric-valued Sobolev maps
本文证明了度量值Sobolev映射的后复合定义在特定空间条件下可仅由定性测试判定，显著推广了既有结果。
Projective freeness and Hermiteness of complex function algebras
本文研究了多种复函数代数的投射自由性与埃尔米特性，提出了极大理想空间上的新上同调充分条件，并基于 Borsuk 形状理论引入新代数类 $\mathscr{C}$，证明其投射张量积能保持这些性质。
The purpose of this paper is to explain a certain dichotomy between the information that the past and future values of a multivariate stochastic process carry about the present. More specifically, vector-valued, second-order stochastic processes may be deterministic in one time-direction and not the other. This phenomenon, which is absent in scalar-valued processes, is deeply rooted in the geometry of the shift-operator. The exposition and the examples we discuss are based on the work of Douglas, Shapiro and Shields on cyclic vectors of the backward shift and relate to classical ideas going ba…
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