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About 2,633 papers for “Vector-valued function”

THE LAPLACE-STIELTJES TRANSFORM OF AN INCREASING VECTOR-VALUED FUNCTION

A. Olubummo · The Quarterly Journal of Mathematics · 1957

THE LAPLACE-STIELTJES TRANSFORM OF AN INCREASING VECTOR-VALUED FUNCTION Get access A. OLUBUMMO A. OLUBUMMO Ibadan Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 8, Issue 1, 1957, Pages 97–107, https://doi.org/10.1093/qmath/8.1.97 Published: 01 January 1957 Article history Received: 25 August 1956 Published: 01 January 1957

Cited by 0

Closed spans in vector-valued function spaces and their approximative properties

А. А. Васильева · Izvestiya Mathematics · 2004

We consider the following fourth-order boundary-value problem:(py )qy = ry, y(0) = y (0) = y (1) = (py )qy (1) + my(1) = 0 with spectral parameter C and physical parameter m R. We assign to this problem a linear pencil of bounded operators Tm = Tm() depending on the physical parameter m and acting from H 2 = y | y W 2 2 [0, 1], y(0) = y (0) = 0 to the dual space H -2 .We study the spectral properties of Tm and use the results of this study to describe properties of the eigenvalues of the problem for various values of m.In particular, we establish asymptotics of these eigenvalues as m 0.

Cited by 12Open access

LIPSCHITZ CONDITIONS FOR THE NORM OF A VECTOR VALUED ANALYTIC FUNCTION

Olivera Djordjevic, Miroslav Pavlović, Communicated Kenneth et al. · 2015

Abstract. We prove some quantitative versions of the Thorp-Whitley max-imum modulus principle as well as extend to vector-valued functions a the-orem of Dyakonov [3] on Lipschitz conditions for the modulus of an analytic functions. Let X be a complex Banach space, and let D be the open unit disk in the complex plane C. Let f: D 7 → X be an analytic function. The function ‖f(·)‖ is subharmonic in D and therefore satisfies the maximum modulus principle: if ‖f(z) ‖ attains its maximum at a point z0 ∈ D, then ‖f(z) ‖ = const. However, the implication (1) ‖f(z) ‖ = const. = ⇒ f(z) = const. need not be true. For example, if X = ` ∞ and f(z) = (1, z, 0,...), then ‖f(z) ‖ ≡ 1. In connection with this problem, Thorp and Whitley [13] introduced the notion of strict c-convexity (or complex strict convexity) as a complex analogue of strict convexity for real Banach spaces. A complex Banach space X is said to be strictly c-convex if for every x, y ∈ X there holds the implication (2) max |z|≤1 ‖x+ zy ‖ ≤ ‖x ‖ = ⇒ y = 0. In other words, X is strictly c-convex if implication (1) holds for complex-linear functions. The result of Thorp and Whitley states the following: 2000 Mathematics Subject Classification. 46B20, 46L52. Key words and phrases. Complex uniform convexity, moduli of c-convexity, Lipschitz condi-tions, maximum principle. The authors are supported by MNZŽS, Grant No. 144010.

Cited by 0

Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action

Dunkl, Charles F. · arXiv (Cornell University) · 2012

The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0}+k_{1}>-\frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $(k_{0},k_{1})$ satisfy $-\frac{1}{2}

Cited by 0Open access

Vector-Valued Multiplier Spaces and Summing Operators: A Modulus Function Approach

Ramazan Kama, Fatma Bulak · Mathematics · 2026

In this paper, we introduce and systematically investigate novel classes of vector-valued multiplier spaces associated with operator-valued series, utilizing the concepts of f-statistical and weak f-statistical convergence. We begin by studying the topological properties of these newly defined spaces, establishing that their completeness is fully characterized by the c0(X)-multiplier convergence of the underlying series. Building upon this structural foundation, we then explore the precise relationships between these f-statistical spaces and classical statistical multiplier spaces, proving that they perfectly coincide under the assumption of a compatible modulus function. Furthermore, we define a natural summing operator acting on these spaces and conduct a detailed analysis of its mapping properties. By establishing necessary and sufficient conditions for the continuity and (weak) compactness of this summing operator, we obtain new characterizations for both c0(X)- and ℓ∞(X)-multiplier convergent series.

Cited by 0Open access

Lipschitz conditions for the norm of a vector valued analytic function

Đorđević, Olivera, Pavlović, Miroslav · Houston journal of mathematics · 2008

We prove some quantitative versions of the Thorp-Whitley maximum modulus principle as well as extend to vector-valued functions a theorem of Dynkonov [3] on Lipschitz conditions For the modulus of an analytic functions.

Cited by 0

A new generalized vector-valued paranormed sequence space using modulus function

Tanweer Jalal, Reyaz Ahmad · Malaya Journal of Matematik · 2015

In this paper we introduce a new generalized vector-valued paranormed sequence spaces $N_p\left(E_k, \triangle_u^m, f, s\right)$ using modulus function $f$, where $p=\left(p_k\right)$ is a bounded sequence of positive real numbers such that $\inf _k p_k>$ $0,\left(E_k, q_k\right)$ is a sequence of seminormed spaces with $E_{k+1} \subseteq E_k$ for each $k \in N$ and $s \geq 0$. We prove results regarding completeness, $K$-space, normality, inclusion relation are derived. These are more general than those of Ruckle [7], Maddox [5], Ozturk and Bilgin [6], Sahiner [8], Atlin et al. [1] and Srivastava and Kumar [9].

Cited by 1Open access

Bounded linear operators on Banach function spaces of vector-valued functions

Neil E. Gretsky, J. J. Uhl · Transactions of the American Mathematical Society · 1972

Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.

Cited by 30Open access

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak · Banach Center Publications · 2005

Let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$. For a real Banach space $(X,\|\cdot\|_X)$ let $E(X)$ be a subspace of the space $L^0(X)$ of $\mu$-equivalence classes of strongly $\Sigma$-measurable functions $f: \O

Cited by 2Open access

Frame multiplication theory and a vector-valued DFT and ambiguity function

Travis D. Andrews, John J. Benedetto, Jeffrey J. Donatelli · arXiv (Cornell University) · 2017

Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory. The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms (STFTs), and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames. In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory.

Cited by 0Open access