This volume contains three articles: "Asymptotic methods in the theory of ordinary differential equations" b'y V. F. Butuzov, A. B. Vasil'eva, and M. V. Fedoryuk, "The theory of best ap proximation in Dormed linear spaces" by A. L. Garkavi, and "Dy namical systems with invariant measure" by A. 'VI. Vershik and S. A. Yuzvinskii. The first article surveys the literature on linear and non linear singular asymptotic problems, in particular, differential equations with a small parameter. The period covered by the survey is primarily 1962-1967. The second article is devoted to the problem of existence, characterization, and uniqueness of best approximations in Banach spaces. One of the chapters also deals with the problem of the convergence of positive operators, inasmuch as the ideas and methods of this theory are close to those of the theory of best ap proximation. The survey covers the literature of the decade 1958-1967. The third article is devoted to a comparatively new and rapid ly growing branch of mathematics which is closely related to many classical and modern mathematical disciplines. A survey is given of results in entropy theory, classical dynamic systems, ergodic theorems, etc. The results surveyed were primarily published during the period 1956-1967.
Inhalt
Asymptotic Methods in the Theory of Ordinary Differential Equations.- I: Linear Differential Equations.- 1. Introduction.- 2. Parameter less Singular Problems.- 3. Problems Regular in x Involving a Parameter.- 4. Problems Containing a Parameter and Singular in x.- II: Nonlinear Differential Equations.- 1. Introduction.- 2. The Cauchy Problem.- 3. Asymptotic Expansions of the Solution of the Cauchy Problem.- 4. Problems with Other Supplementary Conditions.- 5. The Phenomenon of Separation.- 6. Irregularity Governed by an Infinite Range of Variation of the Independent Variable. The Method of Averaging.- 7. Integro-differential Equations.- 8. Systems with a Small Delay.- The Theory of Best Approximation in Normed Linear Spaces.- I. Best Approximation by Polynomials and Their Generalization in Classical Function Spaces.- 1. The Space of Continuous Functions.- 2. Spaces with Integral Metrics.- II. The Problem ol Best Approximation in Banach Spaces.- 1. The Approximation of Continuous Abstract Functions.- 2. Arbitrary Normed Spaces.- 3. Some Special Problems of Best Approximation.- III. Geometric Problems in the Theory of Best Approximation.- 1. The Problem of the Convexity of a Chebyshev Set.- 2. Other Approximative Geometric Problems.- IV. On the Approximation of Sets.- 1. Deviation of Sets.- 2. The Best Approximating Set.- V. On the Convergence of Positive Linear Operators.- Dynamical Systems with Invariant Measure.- 1. Introduction.- 2. Entropy Theory.- 3. Spectral Theory and Operator Rings.- 4. Ergodic Theorems.- 5. Classical Systems.- 6. Systems of Algebraic and Number-Theoretic Extraction.- 7. Transformations with Infinite and Quasi-Invariant Measure and the Existence of an Invariant Measure.- 8. Other Questions.