On the Ergodic Averages and the Ergodic Hilbert Transform

1995 ◽  
Vol 47 (2) ◽  
pp. 330-343
Author(s):  
L. M. Fernández-Cabrera ◽  
F. J. Martín-Reyes ◽  
J. L. Torrea
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Dual Problem ◽  
Finite Measure ◽  
Positive Function ◽  
Ergodic Averages ◽  
Unified Approach ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
Abstract Method

AbstractLet T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

Ergodic Properties of Lamperti Operators, II

1983 ◽  
Vol 35 (4) ◽  
pp. 577-588 ◽  
Author(s):  
Charn-Huen Kan
Keyword(s):  
Positive Operator ◽  
Measure Space ◽  
Finite Measure ◽  
Classical Case ◽  
Variant Form ◽  
Ergodic Properties ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
General Power ◽  
Power Bounded

For T in our main Theorem 5, T* is called Lamperti in [11], whose terminology and notation we shall follow in the sequel. To avoid longish expressions, we shall also say that T* here is disjunctive and, dually, T = (T*)* is codisjunctive. The present work grows out of an attempt to establish a DEE for the general power bounded positive operator on Lp, in view of the success in the contraction case [1, 11], and forms a continuation of [11]. (In passing, we note that Calderon's technique [2] mentioned in [11] was anticipated in 1938 by M. Fukamiya [7], though in a variant form and for a more classical case, namely that of a positive Lp isometry induced by an invertible, measure preserving transformation on a totally finite measure space. Calderon's case does not assume invertibility nor total finiteness.)

A Simple Proof of the Maximal Ergodic Theorem

1976 ◽  
Vol 28 (5) ◽  
pp. 1073-1075 ◽  
Author(s):  
Alberto De La Torre
Keyword(s):  
Ergodic Theorem ◽  
Simple Proof ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Function ◽  
Linear Transformations ◽  
Finite Measure Space ◽  
Image Position

Let X be a σ-finite measure space and let Tk, k any integer, be a group of positive linear transformations in Lp(X) such thatwith C independent of / and k. From now on / will be a positive function in Lp(X) and we will use the following notation:

On the a.s. convergence of the one-sided ergodic Hilbert transform

2009 ◽  
Vol 29 (6) ◽  
pp. 1781-1788 ◽  
Author(s):  
CHRISTOPHE CUNY
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Unitary Operator ◽  
Finite Measure ◽  
Positive Answer ◽  
Positive Contraction ◽  
Finite Measure Space ◽  
The One

AbstractWe show that for T a Dunford–Schwartz operator on a σ-finite measure space (X,Σ,μ) and f∈L1(X,μ), whenever the one-sided ergodic Hilbert transform ∑ n≥1(Tnf/n) converges in norm, it converges μ-a.s. A similar result is obtained for any positive contraction of some fixed Lp(X,Σ,μ), p>1. Applying our result to the case where T is the (unitary) operator induced by a measure-preserving (invertible) transformation, we obtain a positive answer to a question of Gaposhkin.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

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