Topics in Harmonic Analysis and Ergodic Theory.

Intro -- Contents -- Preface -- List of Participants -- Topics in Ergodic Theory and Harmonic Analysis: An Overview -- The mathematical work of Roger Jones -- The Central Limit Theorem for Random Walks on Orbits of Probability Preserving Transformations -- Probability, Ergodic Theory, and Low-Pass Filters -- (1) Introduction. An overview. Basic notation -- (2) Two simple examples: the Haar function and the stretched Haar function. Correcting defective filters -- (3) An outline of the probability argument: Low-pass filters as transition probabilities and a zero-one principle -- (4) The Paul Lévy Borel-Cantelli Lemma and the convergence/divergence of an infinite product -- (5) Doeblin's coupling for low-pass filters -- (6) The state space and the path space. Basic probability theory for this application -- (7) Coding R1 into the state space: The signed magnitude representation versus the two's complement representation -- (8) The construction of a stationary Markov process. P-invariant measures, martingales, and harmonic functions -- (9) The crux of the problem: Invariant sets. Cycles and perfect sets. Forbidden zeros -- (10) The asymptotic behavior of paths from an initial point. Recurrent and transient points. Attractors and inaccessible sets. Examples -- (11) The probabilistic description of low-pass filters (Theorem 11.1) -- (12) The polynomial case: Daubechies' filters and the Pascal-Fermat correspondence. Cohen's necessary and sufficient conditions. A zero-one principle (Theorem 12.1) -- (13) Analytic conditions for low-pass filters. A class of examples from subshifts of finite type (Theorem 13.1) -- (14) Concluding remarks -- (15) References -- Ergodic Theory on Borel Foliations by Rn and Zn -- Short review of the work of Professor J. Marshall Ash -- Uniqueness questions for multiple trigonometric series -- 1. Introduction.

2. Some Cantor-Lebesgue Type Theorems -- 2.1. Square Summation -- 2.2. Restrictedly Rectangular Summation -- 2.3. Unrestrictedly Rectangular Summation -- 2.4. Spherical Summation -- 3. A Uniqueness Theorem for Unrestrictedly Rectangular Convergence -- 4. A Uniqueness Theorem for Spherical Convergence -- 5. Sets of Uniqueness under Spherical Summation -- 6. Questions about Square and Restricted Rectangular Uniqueness -- 6.1. Three weak theorems -- 6.2. Some conjectures -- 6.3. Towards a counterexample -- 7. Orthogonal Trigonometric Polynomials -- References -- Smooth interpolation of functions on Rn -- Problems in interpolation theory related to the almost everywhere convergence of Fourier series -- Lectures on Nehari's Theorem on the Polydisk -- The s-function and the exponential integral.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.