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 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.

A REMARK CONCERNING THE THERMODYNAMICAL LIMIT OF THE LYAPUNOV SPECTRUM

1996 ◽  
Vol 06 (06) ◽  
pp. 1137-1142 ◽  
Author(s):  
Ya. G. SINAI
Keyword(s):  
Dynamical Systems ◽  
Short Range ◽  
Limiting Distribution ◽  
Lyapunov Spectrum ◽  
Short Range Interaction ◽  
Range Interaction ◽  
Limit Transition ◽  
Thermodynamical Limit ◽  
Definition Of ◽  
General Conditions

We consider dynamical systems of N particles confined in domains of volume V with pair-wise short range interaction. We propose a new definition of the limiting distribution of Lyapunov Spectrum according to which the thermodynamical limit transition is taken before the limit t→∞. The main result proven under rather general conditions gives the existence of this modified limiting distribution of Lyapunov Spectrum.

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 Integration Over a Generic Algebra

1997 ◽  
Vol 12 (32) ◽  
pp. 5803-5826 ◽  
Author(s):  
R. Casalbuoni
Keyword(s):  
Path Integral ◽  
Configuration Spaces ◽  
Integral Formalism ◽  
Explicit Algorithm ◽  
Path Integral Formalism ◽  
Generic Algebra ◽  
Definition Of ◽  
General Conditions

In this paper we consider the problem of quantizing theories defined over configuration spaces described by noncommuting parameters. If one tries to do that by generalizing the path-integral formalism, the first problem one has to deal with is the definition of integral over these generalized configuration spaces. This is the problem we state and solve in the present work, by constructing an explicit algorithm for the integration over a generic algebra. The general conditions a given algebra has to satisfy in order to admit our integration are not yet fully understood, but many examples are discussed in order to illustrate our construction.

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.

On extensions of the Riemann and Lebesgue Integrals by Nets

1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Set ◽  
Denjoy Integral ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Open Extension

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

Article 54 Rules on the establishment of the supervisory authority

2020 ◽  
Author(s):  
Hielke Hijmans
Keyword(s):  
Data Protection ◽  
Supervisory Authority ◽  
Definition Of ◽  
General Conditions

Article 4(21) (Definition of a supervisory authority); Article 38 (Position of the data protection officer); Article 51 (Establishment of supervisory authorities); Article 52 (Independence of supervisory authorities) (see too recitals 118–120); Article 53 (General conditions for the members of supervisory authorities); Article 76 (EDPB Confidentiality).

Article 4(21). Supervisory authority

2020 ◽  
Author(s):  
Lee A. Bygrave
Keyword(s):  
Data Protection ◽  
Supervisory Authority ◽  
European Data ◽  
Definition Of ◽  
General Conditions

Article 4(22) (Definition of ‘supervisory authority concerned’); Article 51 (Supervisory authority); Article 52 (Independence) (see too recitals 118 and 120–121); Article 53 (General conditions for the members of the supervisory authority) (see too recital 121); Article 54 (Rules on the establishment of the supervisory authority); Articles 55–59 (Competence, tasks and powers) (see too recitals 122–124, 129 and 132); Articles 60–62 (Cooperation) (see too recitals 125–128, 130–131 and 133–134); Articles 63–67 (Consistency) (see too recitals 119 and 135–138); Article 68 (European Data Protection Board) (see too recital 139).

Article 53 General conditions for the members of the supervisory authority

2020 ◽  
Author(s):  
Hielke Hijmans
Keyword(s):  
Data Protection ◽  
Supervisory Authority ◽  
European Data ◽  
Definition Of ◽  
General Conditions

Article 4(21) (Definition of a supervisory authority); Article 51 (Establishment of supervisory authorities) (see too recital 117); Article 52 (Independence of supervisory authorities) (see too recitals 118–120); Article 54 (Rules on the establishment of supervisory authorities); Article 68 (European Data Protection Board) (see too recital 139).

Sign in / Sign up

Export Citation Format

Share Document