scholarly journals Measure convergent sequences in Lebesgue spaces and Fatou's lemma

1996 ◽  
Vol 54 (2) ◽  
pp. 197-202 ◽  
Author(s):  
Heinz-Albrecht Klei
Keyword(s):  
Integrable Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Integrability ◽  
Lebesgue Spaces ◽  
Integrable Functions ◽  
Fatou’S Lemma ◽  
Necessary And Sufficient ◽  
Fatou's Lemma ◽  
Convergent Sequences

Let (fn) be a sequence of positive P-integrable functions such that (∫ fndP)n converges. We prove that (fn) converges in measure to if and only if equality holds in the generalised Fatou's lemma. Let f∞ be an integrable function such that (∥fn − f∞∥1)n converges. We present in terms of the modulus of uniform integrability of (fn) necessary and sufficient conditions for (fn) to converge in measure to f∞.

scholarly journals Potential Operators on Cones of Nonincreasing Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Alexander Meskhi ◽  
Ghulam Murtaza
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Product Type ◽  
Necessary And Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Potential Operators ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators(Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dtand(ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτon the cone of nonincreasing functions are derived. In the case ofℐα1,α2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.

scholarly journals On the Riesz Almost Convergent Sequences Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Mehmet Şengönül ◽  
Kuddusi Kayaduman
Keyword(s):  
Sequence Space ◽  
Sufficient Conditions ◽  
Riesz Transforms ◽  
Necessary And Sufficient Conditions ◽  
Infinite Matrices ◽  
Necessary And Sufficient ◽  
Convergent Sequences

The purpose of this paper is to introduce new spaces and that consist of all sequences whose Riesz transforms of order one are in the spaces and , respectively. We also show that and are linearly isomorphic to the spaces and , respectively. The and duals of the spaces and are computed. Furthermore, the classes and of infinite matrices are characterized for any given sequence space and determine the necessary and sufficient conditions on a matrix to satisfy , for all .

scholarly journals On bilinear weighted inequalities with Volterra integral operators

2019 ◽  
Vol 486 (4) ◽  
pp. 416-420
Author(s):  
V. D. Stepanov ◽  
G. E. Shambilova
Keyword(s):  
Sufficient Conditions ◽  
Integral Operators ◽  
Necessary And Sufficient Conditions ◽  
Weighted Inequalities ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Necessary And Sufficient

Necessary and sufficient conditions on the boundedness in weighted Lebesgue spaces on the semiaxis for bilinear inequalities with Volterra integral operators are given.

scholarly journals GROUPOID -ALGEBRAS WITH HAUSDORFF SPECTRUM

2013 ◽  
Vol 88 (2) ◽  
pp. 232-242 ◽  
Author(s):  
GEOFF GOEHLE
Keyword(s):  
Haar System ◽  
Sufficient Conditions ◽  
Orbit Space ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Fell Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractSuppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.

scholarly journals On the inclusion of a bounded convergence field in the space of almost convergent sequences

1972 ◽  
Vol 13 (1) ◽  
pp. 82-90 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Sufficient Conditions ◽  
Research Problem ◽  
Necessary And Sufficient Conditions ◽  
General Context ◽  
Regular Matrix ◽  
Reverse Inclusion ◽  
Necessary And Sufficient ◽  
Convergent Sequences

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.

scholarly journals Summability of a Tchebysheff system of functions

2010 ◽  
Vol 8 (1) ◽  
pp. 87-102 ◽  
Author(s):  
Z. T. Abdikalikova ◽  
A. A. Kalybay
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Interest ◽  
Lebesgue Spaces ◽  
Necessary And Sufficient

We consider a special type of Tchebysheff systems of functions{ui(⋅)}in=0and{Vi(⋅)}in=0defined on the intervals (0, 1] and [1,+∞), respectively, such thatui(t)=tα0∫t1t1α1∫t11t2α2…∫ti−11tiαidtidti−1…dt1andui(t)=tβ0∫1tt1β1∫1t1t2β2…∫1ti−1tiβidtidti−1…dt1. We find necessary and sufficient conditions under which functions from the investigated systems belong to the corresponding Lebesgue spacesLp(0, 1) andLp(1,+∞). In order to prove the main results we obtain lower and upper estimates of these functions that are of independent interest.

scholarly journals Generalized Nörlund and Nörlund-type Means of Sequences of Fuzzy Numbers

2020 ◽  
Vol 13 (5) ◽  
pp. 1088-1096
Author(s):  
Pradosh Kumar Pattanaik ◽  
Susanta Kumar Paikray ◽  
Bidu Bhusan Jena
Keyword(s):  
Fuzzy Numbers ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Slow Oscillation ◽  
Real Numbers ◽  
Fuzzy Real Numbers ◽  
Sequences Of Fuzzy Numbers ◽  
Necessary And Sufficient ◽  
Convergent Sequences

In this article we study some properties of generalized Nörlund and Nörlund-typemeans of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for our purposed methods to transform convergent sequences of fuzzy real numbers into convergent sequences of fuzzy real numbers which also preserve the limit. Finally, we establish some results showing the connection between the generalized N ̈orlund and N ̈orlund-type limits and the usual limits under slow oscillation of sequences of fuzzy real numbers.

scholarly journals Matrix transformations from absolutely convergent series to convergent sequences as general weighted mean summability methods

2000 ◽  
Vol 24 (8) ◽  
pp. 533-538
Author(s):  
Jinlu Li
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Convergent Series ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Weighted Mean ◽  
Summability Methods ◽  
Necessary And Sufficient ◽  
Convergent Sequences

We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.

scholarly journals A metrizable semitopological semilattice with non-closed partial order

2020 ◽  
Vol 8 (1) ◽  
pp. 67-75
Author(s):  
Taras Banakh ◽  
Serhii Bardyla ◽  
Alex Ravsky
Keyword(s):  
Partial Order ◽  
Dense Subset ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Countable Family ◽  
Hausdorff Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractWe construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

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