scholarly journals Bounded analytic functions and absolutely continuous measures

1967 ◽  
Vol 18 (5) ◽  
pp. 818-818 ◽  
Author(s):  
George Piranian ◽  
Allen L. Shields ◽  
James H. Wells
Keyword(s):  
Analytic Functions ◽  
Absolutely Continuous ◽  
Bounded Analytic Functions ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

scholarly journals Gibbs measures for partially hyperbolic attractors

1982 ◽  
Vol 2 (3-4) ◽  
pp. 417-438 ◽  
Author(s):  
Ya. B. Pesin ◽  
Ya. G. Sinai
Keyword(s):  
Special Form ◽  
Gibbs Measures ◽  
Hyperbolic Attractor ◽  
Absolutely Continuous ◽  
Unstable Manifolds ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Limit Points ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

AbstractWe consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.

scholarly journals Estimation of integrals with respect to infinite measures using regenerative sequences

2015 ◽  
Vol 52 (04) ◽  
pp. 1133-1145 ◽  
Author(s):  
Krishna B. Athreya ◽  
Vivekananda Roy
Keyword(s):  
Random Walk ◽  
Confidence Interval ◽  
Measure Space ◽  
Integrable Function ◽  
Consistent Estimator ◽  
Absolutely Continuous ◽  
Moment Conditions ◽  
Symmetric Random Walk ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝdusing a simple symmetric random walk on ℤ.

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

2009 ◽  
Vol 19 (01) ◽  
pp. 409-417 ◽  
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measures ◽  
Stochastic Matrix ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Higher Dimension ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Higher Dimensional ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

BAYES' FORMULA FOR SECOND QUANTIZATION OPERATORS

2006 ◽  
Vol 06 (02) ◽  
pp. 245-253 ◽  
Author(s):  
ALBERTO LANCONELLI
Keyword(s):  
Conditional Expectation ◽  
Bayes Rule ◽  
Second Quantization ◽  
Absolutely Continuous ◽  
Bayes Formula ◽  
Conditional Expectations ◽  
Quantization Operator ◽  
Continuous Measures ◽  
The Relationship ◽  
Absolutely Continuous Measures

The Bayes' formula provides the relationship between conditional expectations with respect to absolutely continuous measures. The conditional expectation is in the context of the Wiener space — an example of second quantization operator. In this note we obtain a formula that generalizes the above-mentioned Bayes' rule to general second quantization operators.

A Bound on the Number of Invariant Measures

1981 ◽  
Vol 24 (1) ◽  
pp. 123-124 ◽  
Author(s):  
Abraham Boyarsky
Keyword(s):  
Upper Bound ◽  
Invariant Measures ◽  
Absolutely Continuous ◽  
Continuous Measures ◽  
Absolutely Continuous Measures ◽  
Points Of Discontinuity

For τ a piecewise C2 transformation, we present a method for obtaining an upper bound for the number of independent absolutely continuous measures invariant under τ.Let τ = [0,1] and let τ:I→ J be a piecewise C2 transformation with infI1 |dτ/dx| > 1, where I1 = I-P and P denotes the points of discontinuity of τ and τ′

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