Hausdorff Transforms of Bounded Sequence

1960 ◽  
Vol 11 (1) ◽  
pp. 84
Author(s):  
J. H. Wells
Keyword(s):  
Bounded Sequence

VMO Space Associated with Parabolic Sections and its Application

2015 ◽  
Vol 58 (3) ◽  
pp. 507-518
Author(s):  
Ming-Hsiu Hsu ◽  
Ming-Yi Lee
Keyword(s):  
Hardy Space ◽  
Weak Convergence ◽  
Bounded Sequence ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

AbstractIn this paper we define a space VMO𝒫 associated with a family 𝒫 of parabolic sections and show that the dual of VMO𝒫 is the Hardy space . As an application, we prove that almost everywhere convergence of a bounded sequence in implies weak* convergence

Center Points Of Nets

1975 ◽  
Vol 27 (2) ◽  
pp. 418-422 ◽  
Author(s):  
C. L. Anderson ◽  
W. H. Hyams ◽  
C. K. McKnight
Keyword(s):  
Metric Space ◽  
Limit Point ◽  
Cauchy Sequence ◽  
Bounded Sequence ◽  
Center Point

Suppose x = (x∝) is a net with values in a metric space X having metric ρ. If a point z in X can be found to minimizethen z is called a center point (c.p.) of x. The space X is (netwise) c.p. complete if every bounded net has at least one c.p.; it is sequentially c.p. complete if every bounded sequence has a c.p. Netwise c.p. completeness implies sequential c.p. completeness, and the latter implies completeness since any c.p. of a Cauchy sequence will necessarily be a limit point of that sequence.These notions are related to the set centers of Calder et al. [2].

scholarly journals Axiomatisations of the average and a further generalisation of monotonic sequences

1974 ◽  
Vol 15 (1) ◽  
pp. 63-65 ◽  
Author(s):  
John Bibby
Keyword(s):  
Bounded Sequence ◽  
Monotonic Sequence

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

scholarly journals Kadec–Pełczyński decomposition for Haagerup Lp-spaces

2002 ◽  
Vol 132 (1) ◽  
pp. 137-154 ◽  
Author(s):  
NARCISSE RANDRIANANTOANINA
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Bounded Sequence ◽  
Point Property ◽  
Weakly Compact ◽  
Von Neumann ◽  
Lp Spaces ◽  
Reflexive Subspace ◽  
Finite Von Neumann Algebras ◽  
Bounded Sequences

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

scholarly journals Statistical limit point theorems

2000 ◽  
Vol 23 (11) ◽  
pp. 741-752 ◽  
Author(s):  
Jeff Zeager
Keyword(s):  
Complex Number ◽  
Limit Point ◽  
Statistical Convergence ◽  
Nonnegative Matrix ◽  
Bounded Sequence ◽  
Regular Matrix ◽  
Number Sequence ◽  
Statistical Limit ◽  
Limit Points

It is known that given a regular matrixAand a bounded sequencexthere is a subsequence (respectively, rearrangement, stretching)yofxsuch that the set of limit points ofAyincludes the set of limit points ofx. Using the notion of a statistical limit point, we establish statistical convergence analogues to these results by proving that every complex number sequencexhas a subsequence (respectively, rearrangement, stretching)ysuch that every limit point ofxis a statistical limit point ofy. We then extend our results to the more generalA-statistical convergence, in whichAis an arbitrary nonnegative matrix.

Asymptotic centres of filters on uniformly rotund spaces

1976 ◽  
Vol 15 (1) ◽  
pp. 87-96
Author(s):  
John Staples
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Bounded Sequence ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Filter Base

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

Absolute summability functions for Hausdorff methods

1982 ◽  
Vol 91 (1) ◽  
pp. 51-56
Author(s):  
B. Kuttner
Keyword(s):  
Absolute Summability ◽  
Bounded Sequence ◽  
Summability Method ◽  
Negative Integer ◽  
Hausdorff Method ◽  
The Matrix ◽  
Sequence Transformation ◽  
Image Position

1. Following Lorentz, we suppose throughout that Ω(n) is a non-negati ve non-decreasing function of the non-negative integer n such that Ω(n)→ ∞ as n → ∞. Consider the summability method given by the sequence-to-sequence transformation corresponding to the matrix A = (ank). We say that Ω(n) is a summability function for A (or absolute summability function for A) if the following holds: Any bounded sequence {sn} such that the number of values of ν with ν ≤ n, sν ≠ 0 does not exceed Ω(n) is summable A (or is absolutely summable A, respectively). These definitions are due to Lorentz (4), (6). We shall be concerned with the case in which A is a regular Hausdorff method, say A = H = (H, μn). Then H is given by the matrix (hnk) withwithX(0) = X(0 + ) = 0, X(1) = 1;(see e.g.(1), chapter XI). We shall suppose throughout that these conditions are satisfied. It is known that H is then necessarily also absolutely regular.

scholarly journals Projection Methods for Ill-Posed Problems Revisited

2016 ◽  
Vol 16 (2) ◽  
pp. 257-276 ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Global Convergence ◽  
Least Squares ◽  
Local Convergence ◽  
Sufficient Conditions ◽  
Projection Methods ◽  
Bounded Sequence ◽  
Oblique Projection ◽  
Ill Posed

AbstractWe consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

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