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.

Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

1995 ◽  
Vol 47 (4) ◽  
pp. 852-876
Author(s):  
David I. McIntosh
Keyword(s):  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Linear Transformations ◽  
Borel Sets ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Sets

AbstractLet ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

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).

Unique Extension and Product Measures

1967 ◽  
Vol 19 ◽  
pp. 757-763 ◽  
Author(s):  
Norman Y. Luther
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Product Measure ◽  
Unique Extension ◽  
Product Measures ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Finite Measures

Following (2) we say that a measure μ on a ring is semifinite ifClearly every σ-finite measure is semifinite, but the converse fails.In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

scholarly journals Interchanging a limit and an integral: necessary and sufficient conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Takashi Kamihigashi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Pointwise Limit ◽  
Necessary And Sufficient

AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

Two-weight weak and extra-weak type inequalities for the one-sided maximal operator

1993 ◽  
Vol 123 (6) ◽  
pp. 1109-1118
Author(s):  
Pedro Ortega Salvador ◽  
Luboš Pick
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Orlicz Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weight Functions ◽  
The One ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLet be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak typeand of extra-weak typehold. Our proofs do not refer to the theory of Orlicz spaces.

scholarly journals Construction of Weights for Positive Integral Operators

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1004 ◽  
Author(s):  
Ron Kerman
Keyword(s):  
Weight Function ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Fundamental Result ◽  
Averaging Operators ◽  
Measurable Functions ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Schur's Lemma

Let ( X , M , μ ) be a σ -finite measure space and denote by P ( X ) the μ -measurable functions f : X → [ 0 , ∞ ] , f < ∞ μ ae. Suppose K : X × X → [ 0 , ∞ ) is μ × μ -measurable and define the mutually transposed operators T and T ′ on P ( X ) by ( T f ) ( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) and ( T ′ g ) ( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X . Our interest is in inequalities involving a fixed (weight) function w ∈ P ( X ) and an index p ∈ ( 1 , ∞ ) such that: (*): ∫ X [ w ( x ) ( T f ) ( x ) ] p d μ ( x ) ≲ C ∫ X [ w ( y ) f ( y ) ] p d μ ( y ) . The constant C > 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form ϕ 1 1 / p − 1 ϕ 2 1 / p , where ϕ 1 , ϕ 2 ∈ P ( X ) satisfy T ϕ 1 ≤ C 1 ϕ 1 and T ′ ϕ 2 ≤ C 2 ϕ 2 . Our fundamental result shows that the ϕ 1 and ϕ 2 above are within constant multiples of (**): ψ 1 + ∑ j = 1 ∞ E − j T ( j ) ψ 1 and ψ 2 + ∑ j = 1 ∞ E − j T ′ ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E > 1 and T ( j ) , T ′ ( j ) are the jth iterates of T and T ′ . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T ′ = T . This means that only the first series in (**) needs to be studied.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

scholarly journals ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

2009 ◽  
Vol 09 (04) ◽  
pp. 635-655 ◽  
Author(s):  
H. BRUIN ◽  
M. NICOL ◽  
D. TERHESIU
Keyword(s):  
Dynamical System ◽  
Common Factor ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Original System ◽  
Return Time ◽  
Tail Behavior ◽  
Necessary And Sufficient ◽  
Young Towers ◽  
Measure Preserving Transformations

For a σ-finite measure preserving dynamical system (X, μ, T), we formulate necessary and sufficient conditions for a Young tower (Δ, ν, F) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling–Kac theorem carries over from F to T. One advantage of the Darling–Kac theorem in terms of Young towers is that sufficient conditions can be read off from the tail behavior and we illustrate this with relevant examples. Furthermore, any two Young towers with a common factor T, have return time distributions with tails of the same order.

Ratio and Stochastic Ergodic Theorems for Superadditive Processes

1979 ◽  
Vol 31 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Humphrey Fong
Keyword(s):  
Linear Operator ◽  
Measure Space ◽  
Measure Zero ◽  
Positive Linear Operator ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Finite Measure Space ◽  
Image Position

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)

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