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 ...

We develop a general framework for the analysis of algorithms of a broad Euclidean type. The average-case 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 average-case 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 gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. 1

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 average-case 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 gcd-like 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.

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.

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.

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Three boundary Nevanlinna-Pick 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.

Abstract. This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operator-theoretic aspects. The term “Holomorphic Spaces, ” the title of a program held at the Mathematical Sciences Research Institute in the fall semester of 1995, is short for “Spaces of Holomorphic Functions. ” It refers not so much to a branch of mathematics as to a common thread running through much of modern analysis—through functional analysis, operator theory, harmonic analysis, and, of course, complex analysis. This article will briefly outline the development of the subject from its origins in the early 1900’s to the present, with a bias toward operator-theoretic aspects, in keeping with the main emphasis of the MSRI program. I hope that the article will be accessible not only to workers in the field but to analysts in general. Origins The subject began with the thesis of P. Fatou [1906], a student of H. Lebesgue.

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 one-dimensional Hausdor# measure. Our work is inspired by an earlier result of Matache [14], who showed that the spaces of half-planes 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".