دامنه و تاریخ تحلیل هارمونیک جابجایی‌پذیر و غیرجابجایی‌پذیر

When I was invited to speak at the conference on the history of analysis given at Rice University [in 1977], I decided that it might be interesting to review the history of mathematics and physics in the last three hundred years or so with heavy emphasis on those parts in which harmonic analysis had played a decisive or at least a major role. I was pleased and somewhat astonished to find how much of both subjects could be included under this rubric ...The picture that gradually emerged as the various details fell into place was one that I found very beautiful, and the process of seeing it do so left me in an almost constant state of euphoria. I would like to believe that others can be led to see this picture by reading my paper, and to facilitate this I have included a large number of short expositions of topics which are not widely understood by non-specialists. --from the Preface This volume, containing the paper mentioned above as well as five other reprinted papers by Mackey, presents a sweeping view of the importance, utility, and beauty of harmonic analysis and its connections to other areas of mathematics and science. A seventh paper, written exclusively for this volume, attempts to unify certain themes that emerged after major discoveries in 1967 and 1968 in the areas of Lie algebras, strong interaction physics, statistical mechanics, and nonlinear partial differential equations--discoveries that may at first glance appear to be independent, but which are in fact deeply interrelated. Information for our distributors: Copublished with the London Mathematical Society beginning with volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. Contents Introduction Harmonic Analysis as the Exploitation of Symmetry: A Historical Survey, Bull. (New Ser.) Amer. Math. Soc. 3 (1980), 543-699. Preface 1. Introduction 2. The Characters of Finite Groups and the Connection with Fourier Analysis 3. Probability Theory Before the Twentieth Century 4. The Method of Generating Functions in Probability Theory 5. Number Theory Before 1801 6. The Work of Gauss and Dirichlet and the Introduction of Characters and Harmonic Analysis into Number Theory 7. Mathematical Physics Before 1807 8. The Work of Fourier, Poisson, and Cauchy, and Early Applications of Harmonic Analysis to Physics 9. Harmonic Analysis, Solutions by Definite Integrals, and the Theory of Functions of a Complex Variable 10. Elliptic Functions and Early Applications of the Theory of Functions of a Complex Variable to Number Theory 11. The Emergence of the Group Concept 12. Introduction to Sections 13-16 13. Thermodynamics, Atoms, Statistical Mechanics, and the Old Quantum Theory 14. The Lebesgue Integral, Integral Equations, and the Development of Real and Abstract Analysis 15. Group Representations and Their Characters 16. Group Representations in Hilbert Space and the Discovery of Quantum Mechanics 17. The Development of the Theory of Unitary Group Representations Between 1930 and 1945 18. Harmonic Analysis in Probability; Ergodic Theory and the Generalized Harmonic Analysis of Norbert Wiener 19. Early Application of Group Representations to Number Theory -- The Work of Artin and Hecke 20. ldèles, Adèles, and Applications of Pontrjagin-van Kampen Duality to Number Theory, Connections with Almost-Periodic Functions, and the Work of Hardy and Littlewood 21. The Development of the Theory of Unitary Group Representations after 1945 -- A Brief Sketch with Emphasis on the First Decade 22. Applications of the General Theory 23. Summary and Conclusion Notes Remarks Bibliography Herman Weyl and the Application of Group Theory to Quantum Mechanics The Significance of Invariant Measures for Harmonic Analysis, Colloquia Mathematica Societatis JANOS BOLYAI 49, Alfred Haar Memorial Conference, Budapest (Hungary), 1985, 551-609 Weyl's Program and Modern Physics, K. Bleuler and M. Werner (eds.), Differential Geometrical Methods in Theoretical Physics, 1988, 11-36. Induced Representations and the Applications of Harmonic Analysis Von Neumann and the Early Days of Ergodic Theory 1. Background. 2. The years 1931 and 1932. 3. Afterwards. Final Remarks 1. Introduction. 2. Roots and Weights in the Theory of Lie Algebras. 3. The Concept of an S Operator. 4. Lattice Statistics and Onsager's Solution of the Ising Problem. 5. Kac-Moody Lie Algebras. 6. The Veneziano Model in Strong Interaction Physics. 7. The Exact Solvability of tlte Korteweg De Vries Equation. 8. The Exact Solvability of the Two-Dimensional "Ice Problem" by Lieb. 10. The Immediate Aftermath. 11. Further Connections Between §5, §6, §7, and §8. 12. Conclusions.