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HARMONIC ANALYSIS AS the EXPLOITATION of SYMMETRY-A HISTORICAL SURVEY Preface 1. Introduction 2. the Characters of Finite Groups

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HARMONIC ANALYSIS AS the EXPLOITATION of SYMMETRY-A HISTORICAL SURVEY Preface 1. Introduction 2. the Characters of Finite Groups BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY-A HISTORICAL SURVEY BY GEORGE W. MACKEY CONTENTS 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 Be­ tween 1930 and 1945 Reprinted from Rice University Studies (Volume 64, Numbers 2 and 3, Spring- Summer 1978, pages 73 to 228), with the permission of the publisher. 1980 Mathematics Subject Classification. Primary 01,10,12, 20, 22, 26, 28, 30, 35, 40, 42, 43, 45,46, 47, 60, 62, 70, 76, 78, 80, 81, 82. Copyright 1978 by Rice University 543 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 544 G. W. MACKEY 18. Harmonie Analysis in Probability; Ergodic Theory and the Generalized Harmonie Analysis of Norbert Wiener 19. Early Application of Group Representations to Number Theory—The Work of Artin and Hecke 20. Idèles, 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 af­ ter 1945—A Brief Sketch with Emphasis on the First Decade 22. Applications of the General Theory 23. Summary and Conclusion Notes Bibliography PREFACE This paper is an expansion (by a factor of twelve) of two talks that I gave in the spring of 1977 at the Rice University Conference on the history of analysis. I am not a historian in the usual professional sense of the word and I did not pretend to be reporting on the results of a careful scholarly investi­ gation. My talks consisted rather of an informal account of the knowledge and impressions I have gained over the years as I have tried to satisfy my curiosity about the origins and interrelationships of the parts of mathemat­ ics and science which most interest me. Moreover, in contradistinction to most (if not all) professional historians of science and mathematics, I was much more interested in getting an overall approximate idea of how our present understanding unfolded than in studying the fine structure of par­ ticular discoveries. I totally ignored such questions as false starts, the thought processes of individual scientists, and the intellectual climate of the times. The question constantly in my mind was this: How much of what we understand now had they grasped by then? When I began in early August 1977 to concentrate on the job of writing up my talks for publication, I found that I could not say what I wanted to say without making a number of assertions about matters of fact that were not always easy to verify but whose exactitude was at most marginally rele­ vant to the story I was trying to tell. I made some attempt to make only cor­ rect assertions (sometimes by being deliberately vague) but my time and pa­ tience were limited and I am sure I did not always succeed. The length of the finished product is not all due to striving for local his­ torical accuracy (this would require at least fifty books), but rather to my desire to make clear the relationship of a large part of mathematics to my License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HARMONIC ANALYSIS AS EXPLOITATION OF SYMMETRY 545 main theme. To this end I composed a large number of brief introductory expositions, which I hope the average mathematician will find intelligible. It is these expositions and their connectedness in time and intellectual content which is the real point of the paper.l 1. INTRODUCTION In this article I shall sketch the history, applications, and ramifications of a certain method. For want of a better word I shall call it the method of har­ monic analysis, although I am aware that many people use these words to denote a different class of generalizations of the classical harmonic analysis of Fourier. In crude terms the method may be described as follows: Let S be a "space" or "set" and let G be a group of one-to-one transformations of S onto itself. Let [s]x denote the transform of s in S by x in G. Ordinarily S will have further structure which will be preserved by the transformations of G so that the transformations s —• [s]x are symmetries of S. It will be con­ venient to allow members of G other than the identity e to define the identi­ ty map so that some quotient group G/N is the actual transformation group. Now let 3 be some vector space of complex valued functions on S which is G invariant in the sense that s — f([s]x), the translate of ƒ by JC, is in S whenever ƒ is in 3. Then for each x in G, the mapping ƒ — g where g(s) = AUM is a linear transformation Vx of 3 onto 3 and V^ = VxVy for all* and y in G. The mapping x — Vx is thus an example of what is called a (lin­ ear) representation of the group G. More generally, a (linear) representation of a group G is by definition any homomorphism x — Wx of G into the group of all bijective linear transformations of some vector space *p (W). The method I propose to discuss in this article consists (in its simplest form) in attempting to find subspaces Mx of the space 3 such that (1) VX(MX) = Mx for all x and X. (2) Every element ƒ in 3 is uniquely a finite or infinite sum ƒ = Lfx where each/xeMx. (3) The subspaces Mx are either not susceptible of further decomposition or are somehow much simpler in structure than 3. Of course, one must have a topology in *J in order to make sense of infinite sums. More generally one also considers "continuous direct sums" or direct integrals and vector- valued as well as complex-valued functions. Of course, each Mx is the space of a new representation ï^, which is a so-called subrepresentation, and one speaks of the direct sum or direct integral decomposition of V. It turns out that the decomposition of functions in 3 into sums and integrals of func­ tions associated with the components of F is a decomposition that greatly simplifies many problems. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 546 G. W. MACKEY 2. THE CHARACTERS OF FINITE COMMUTATIVE GROUPS AND THE CONNECTION WITH FOURIER ANALYSIS Let Wbc a (linear) representation of the group G. If $>(W) is one-dimen­ sional then each Wx is some complex number x(x) times the identity, and x —x(x)I is a representation of G if and only if x(xy) = x(*)xÜ0 for all JC and y in G. When G is a finite commutative group, such functions x are called characters. It is easy to see that the representations x -* x(*) are the only representations of G that are irreducible in the sense that no proper subrep- resentations exist. Let 7i (G) be the vector space of all complex- valued func­ tions on G and let VJXy) = f(yx). Then Kis a representation of G such that £>(K) = S(G) and the one-dimensional subspace generated by each character is clearly an invariant subspace. Indeed every one-dimensional in­ variant subspace is of this form, and one shows easily that Kis a direct sum of one-dimensional representations each associated with a distinct character X. Thus every member ƒ of 3 is uniquely of the form EL cxX(x) X€G where G denotes the set of all characters on G. Evidently the product of two characters is again such, and the set G of all characters is itself a finite com­ mutative group. Since x(x") = x(*)n = 1 when n is the order of x, it follows that | x(x) | = 1 for all x and all x, so x~* = x"for all x« Consider £x(*). This xeG sum is clearly equal to ExOoO = XÖ0 ^>x(x). Thus whenever xOO ^ 1, we xeG xeG have Ex(x) = 0. It follows that £xi(*)x2(*) = 0 whenever xi ^ X2, so that xeG xeG the characters of G are orthogonal with respect to the inner product f*g- £ƒ(*)£(*). xeG Now consider the expansion formula (2.1) fix) = £ CxX(x) X€G Multiplying each side by x '(*) summing over G and using the orthogonality of the characters leads at once to a formula for cx '. (2.2) CX=-JG) E^)XW v ' xeG where o(G) is the number of elements in G.
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