scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

scholarly journals О максимальном квазинормированном расширении квазинормированных векторных решеток

2017 ◽  
Author(s):  
A.G. Kusraev ◽  
B.B. Tasoev
Keyword(s):  
Banach Lattice ◽  
Normed Space ◽  
Fatou Property ◽  
Normed Lattice ◽  
Integrable Functions ◽  
Definition Of

The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension $X^\varkappa$ of a Dedekind complete quasi-normed lattice $X$ with the weak $\sigma$-Fatou property is a quasi-Banach lattice if and only if $X$ is intervally complete. Moreover, $X^\varkappa$ has the Fatou and the Levi property provided that $X$ is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.

scholarly journals Köthe dual of Banach lattices generated by vector measures

2013 ◽  
Vol 173 (4) ◽  
pp. 541-557 ◽  
Author(s):  
Mieczysław Mastyło ◽  
Enrique A. Sánchez-Pérez
Keyword(s):  
Banach Lattices ◽  
Vector Measures ◽  
Köthe Dual

scholarly journals On Pointwise Product Vector Measure Duality

2021 ◽  
Vol 20 ◽  
pp. 8-18
Author(s):  
Levi Otanga Olwamba ◽  
Maurice Oduor
Keyword(s):  
Integral Representation ◽  
Function Space ◽  
Hilbert Function ◽  
Vector Measure ◽  
Inner Product ◽  
Vector Measures ◽  
Integrable Functions ◽  
Space Of Integrable Functions ◽  
Hilbert Function Space ◽  
Product Vector

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

scholarly journals Order and norm convergence in Banach lattices

1974 ◽  
Vol 15 (1) ◽  
pp. 13-13 ◽  
Author(s):  
Andrew Wirth
Keyword(s):  
Banach Lattice ◽  
Banach Lattices ◽  
Norm Convergence ◽  
Order Convergence ◽  
Definition Of

Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

On the Generality of the AP-Integral

1971 ◽  
Vol 23 (3) ◽  
pp. 557-561 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Denjoy Integral ◽  
Integrable Functions ◽  
Definition Of ◽  
General Conditions

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

scholarly journals On the Inversion of the Gauss Transformation

1957 ◽  
Vol 9 ◽  
pp. 459-464 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Differential Operator ◽  
Recent Work ◽  
Suitable Definition ◽  
Inversion Theory ◽  
The Subject ◽  
Definition Of ◽  
Image Position ◽  
Gauss Transformation ◽  
Operational Methods

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by1.1,then operational methods indicate that,under a suitable definition of the differential operator.

Insensitive average residence times in generalized semi-Markov processes

1981 ◽  
Vol 13 (04) ◽  
pp. 720-735 ◽  
Author(s):  
A. D. Barbour ◽  
R. Schassberger
Keyword(s):  
Steady State ◽  
Markov Processes ◽  
Broad Class ◽  
Residence Times ◽  
Steady State Distribution ◽  
State Distribution ◽  
Lifetime Distributions ◽  
Definition Of ◽  
Semi Markov Processes ◽  
Networks Of Queues

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

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