Results 1  10
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73
Dynamical Sources in Information Theory: Fundamental intervals and Word Prefixes.
, 1998
"... A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewe ..."
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Cited by 28 (7 self)
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A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a "generating" operator. Its dominant spectral objects are linked with important parameters of the source such as the entropy, and play a central role in all the results. 1 Introduction. In information theory contexts, data items are (infinite) words that are produced by a common mechanism, called a source. Realistic sources are often complex objects. We work here inside a quite general framework of sources related to dynamical systems theory which goes beyond the cases of memoryless and Markov sources. This model can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, they attain a high level of generality. A probabilistic dynamical source ...
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 17 (4 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Continued Fractions, Comparison Algorithms, and Fine Structure Constants
, 2000
"... There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems ..."
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Cited by 10 (2 self)
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There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.
Hardy Spaces That Support No Compact Composition Operators
 J. FUNCTIONAL ANALYSIS
, 2003
"... We consider, for G a simply connected domain and 0 < p < (G) formed by fixing a Riemann map # of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of over the curves #({z = r}) be bounded for 0 < r < 1. The resulting space is usually not the one obtai ..."
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Cited by 7 (2 self)
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We consider, for G a simply connected domain and 0 < p < (G) formed by fixing a Riemann map # of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of over the curves #({z = r}) be bounded for 0 < r < 1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: (G) supports compact composition operators if and only if #G has finite onedimensional Hausdor# measure. Our work is inspired by an earlier result of Matache [14], who showed that the spaces of halfplanes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with "compact" replaced by "Riesz".
Composition operators between Bergman and Hardy spaces
 Trans. Amer. Math. Soc
, 1996
"... Abstract. We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results ..."
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Cited by 6 (4 self)
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Abstract. We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces. 1.
On the NevanlinnaPick interpolation problem for generalized Stieltjes functions, Integral Equations Operator Theory 30
, 1998
"... Three boundary NevanlinnaPick interpolation problems at finitely many points are formulated for generalized Schur functions. For each problem, the set of all solutions is parametrized in terms of a linear fractional transformation with a Schur class parameter. 1. ..."
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Cited by 6 (3 self)
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Three boundary NevanlinnaPick interpolation problems at finitely many points are formulated for generalized Schur functions. For each problem, the set of all solutions is parametrized in terms of a linear fractional transformation with a Schur class parameter. 1.
Backwarditeration sequences with bounded hyperbolic steps for analytic selfmaps of the disk
 Rev. Mat. Iberoamericana
"... Abstract. A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The DenjoyWolff Theorem describes their convergence properties and several authors, from the 1880’s to the 1980’s, have provided conjugations which yield very precise des ..."
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Cited by 6 (4 self)
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Abstract. A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The DenjoyWolff Theorem describes their convergence properties and several authors, from the 1880’s to the 1980’s, have provided conjugations which yield very precise descriptions of the dynamics. Backwarditeration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forwarditeration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backwarditeration sequences with bounded hyperbolic steps for analytic selfmaps of the disk. 1.