DOCSLIB.ORG

1 Introduction 2 Haar Measure Z

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Introduction 2 Haar Measure Z Harmonic Analysis EPvan den Ban Master Class Course MRI Fall Intro duction For an intro duction to the following material we refer to the rst lecture given by the author One may also confer Sug which will b e the basic reference for the rest of this course Haar measure Let X b e a lo cally compact Hausdor space Denition A p ositive Radon integral on X is a linear functional I C X C such that c f I f Remark By the Riesz representation theorem I is actually the integral asso ciated with a regular Borel measure We shall refer to this measure as the Radon measure asso ciated with I and we shall use the notation Z I f f x dx G Lemma Let I b e a p ositive Radon measure on X Then a f g I f I g f g C X c b f C X jI f j I jf j c c The map f I f is continuous on C G c Proof For a use that g f and the denition of I b First assume that f is realvalued and dene continuous functions by f maxf and f maxf Then f f f and jf j f f Hence jI f j jI f I f jI f I f I jf j Now assume that f is complex valued and that I f R Then I Im f Hence jI f j jI Re f j I jRe f j I jf j Sug M Sugiura Unitary Representations and Harmonic Analysis nd ed NorthHollandKo ddans ha Amsterdam by what we proved ab ove Let f now b e arbitrary Then there exists a z C with jz j and zI f Then I zf R hence jI f j jI zf j I jzf j I jf j It remains to establish c Let K X b e a compact subset Then wemust show that there exists a C such that for all f C K wehave jI f j C kf k There exists a nonnegative c function g C X such that g on K Hence for all f C K wehave jf j jf jg kf k g c c and it follows that jI f j I jf j kf k I g which establishes the desired estimate Note that by taking the inmum over g one obtains the estimate jI f j kf k volume K By a Gspace we shall from now on mean a lo cally compact Hausdor space X together with a continuous action of G on X We agree to write L for the induced action of G on C X Thus if C X then L x g g x The Gspace X is called homogeneous if there is only one Gorbit ie if x X is a p oint then G x X One readily veries that in this setting the stabilizer G fg G gx x g x is a closed subgroup of G Moreover the map G X g gx induces a homeomorphism GG X Thus any homogeneous Gspace is of the form GH with H a closed subgroup of x G ConverselyifH is a closed subgroup then GH is a homogeneous space in a natural way Example The unit sphere S in R is a homogeneous space for the sp ecial orthogonal group SO Let e Then the stabilizer of e in SO is the group of rotations with axis Re which is isomorphic to SO The sphere is thus identied with the quotientSOSO This is the starting p oint for the spherical harmonics to b e discussed later Lemma Let X be a Gspace and let C X Then g L is a continuous map c g G C X c Proof Let g G Then wemust show that L L is close to in C X for g close to g g c where L and we see that without loss of generality w L L L g No g g g gg wemayaswell assume that g e Let C b e a compact neighb ourho o d of e in G Then the set K C supp is compact in X Moreover for g C the function L has supp ort contained g in K Fix Then wemust show that there exists an op en neighb ourho o d U of e in C such that jL kL k sup x xj g g K xK for all x U The map g x g xiscontinuous hence for every x K there exists an op en neighb ourho o d N x in X and an op en neighb ourho o d U e in C G such that x x jg y y j for all y N g U By compactness of K there exists a nite collection x x x x of p oints in K such that the sets N cover K Let U be the intersection of the sets n x j U Then U is an op en neighb ourho o d of e in G and one readily checks that holds for g U j Corollary Let X be a Gspace and let I b e a p ositive Radon integral on G Then for C X wehavethatg I L denes a continuous function G C c g A p ositive Radon integral I on a Gspace X is called invariant if for all C X and c g G wehave I L I g or equivalentlyif denotes the measure asso ciated with I Z Z g xdx x dx X X Lemma Let I be a Ginvariant p ositive nontrivial Radon measure on a Ghomogeneous space X Let f C X and supp ose that f everywhere Then I f f c Proof Supp ose that f is as ab ove and I f but f Then there exists a p oint x X such that f x Hence there exists a compact neighb ourho o d U x and a p ositve constant Let C suppg b e the compact supp ort such that f on U Let now g C X g c of g Then every p ointofC is contained in Gx and by compactness there exists a nite subset S G such that the sets sU s S cover C For every s S there exists a such that on s P sU wehave g L f Hence g L f and it follows that s s s s s sS X I g I L f s s sS tegral It follows from this the latter equality b eing a consequnece of the invariance of the in that I contradiction Corollary Let the hyp othesis of the ab ove lemma b e fullled Then Z hf jg i I f g f xg x dx X denes a p ositive denite Hermitian inner pro duct on C X c We denote the completion of C X with resp ect to the ab ove inner pro duct by L X Let c L denote the left represenation of G on X Then byinvariance of the measure the map L is x unitary for the inner pro duct on C x and therefore has a unique extension to a unitary map c L L X L X x The group G may b e viewed as a homogeneous space for its left right action A left right invariantpositive nontrivial Radon integral on X is called a left right Haar integral on G Theorem Let G b e a lo cally compact group Then there exists a left right Haar integral I on G If I is a second left right Haar integral on G then I cI for a p ositive constant c Proof For the existence part of this theorem we refer the reader to BI We will prove uniqueness If f C X then we dene the function f C X by f xf x If J is a c c BI Bourbaki Integration Chapter right Haar integral then we dene the left Haar integral J by J f J f Thus wehavea bijection b etween left and rightHaarintegrals Let now I and J b e a left and a right Haar integral resp ectively Then it suces to show that J cI for some c Let and denote the measures asso ciated with I and J resp ectively and let f C Gbesuch that I f Dene c Z D x I f f y x d y f G Then D is a continuous function on Gby Lemma f Let g C G Then c Z J g g xy d x G for every y G Multiplying the ab ove identity with f y andintegrating with resp ect to dy we obtain Z Z Z Z I f J g f y g xy d xdy f y g xy dy d x G G G G where the change of order of integration is allowed byFubinis therorem Note that the integrand is compactly supp orted as a function on G G Using left invariance of and after that again changing the order of integration wemay rewrite the integral as Z Z f x y d x g y dy f D g f G G Hence J g I D g for every g C GLet f f C Gbetwo functions with I f f c c j ws from the ab ove that I D D g for all g hence D D Thus it then it follo F f f f follows that D D is a continuous function which is actually indep endent of the particular f choice of f Substituting x e in the denition of D wenow infer that f D eI f J f for all f in the set S of functions C GwithI The set S is readily seen to b e dense c in C G so that is actually valid for all f C G Hence J D eI c c Remark The existence and the uniqueness of Haar measure is much easier to establish when G is a Lie group ie G p ossesses the structure of C manifold compatible with its top ologyandsuch that the map G G G x y xy is smo oth ie C Examples of Lie groups are SUn SOn SLn C GLn C SLn R GLn R For simplicitywe assume n denotes that G is connected Let n dim G and x a nontrivial nform T G Here T G e e x the cotangentspace of G at an element x G Let l G G g xg and dene for x G x n the nform T G by x x T l x e x e Then x is a nowhere vanishing smo oth dierential nform on G hence denes an x orientation on G We dene I C G C by the oriented integral c Z I f f f C G c G One readily veries that I is a p ositiveRadonintegral which is nontrivial Moreover one also readily checks that l for all g G from which it follows that I is a left Haar integral g For further details we refer to BD or Wa Let I b e a left Haar integral on G Then for every x G we dene the Radon integral R I x by R I I R Then one readily veres that R I isaleftHaarintegral use that R x x x x and L commute Hence R I xI for a uniquely determined x The function y x G which is readily seen to b e indep endentoftyhe particular choice of I is called the mo dular function of G Exercise Show that is a continuous group homomorphism A group G with is called unimo dular Thus a group is unimo dular if its left Haar measure is a right Haar measure For obvious reasons the Haar measure is said to b e biinvariant in this case Exercise Show that a compact group G is unimo dular Hint use that the image G is a compact group Representations From nowonG will always b e a lo cally compact top ological group Exercise Let b e a representation of G in a Banach space V Show that the following conditions are equivalent a G V V is continuous b For every x G the map x is coninuous and for every v V the map G V x x v is continuous Hint to show that a follows from b use the BanachSteinhaus theorem Lemma Let X be a Ghomogeneous space and let I b e a nontrivial invariant p ositive Radon integral on X Then the representation L of G in L X dened in the previous section is continuous Proof In view of
Load more

Recommended publications
  • Geometric Integration Theory Contents
    Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants .
    [Show full text]
  • Measure Theory John K. Hunter
    Measure Theory John K. Hunter Department of Mathematics, University of California at Davis Abstract. These are some brief notes on measure theory, concentrating on n Lebesgue measure on R . Some missing topics I would have liked to have in- cluded had time permitted are: the change of variable formula for the Lebesgue n integral on R ; absolutely continuous functions and functions of bounded vari- ation of a single variable and their connection with Lebesgue-Stieltjes measures n on R; Radon measures on R , and other locally compact Hausdorff topological spaces, and the Riesz representation theorem for bounded linear functionals on spaces of continuous functions; and other examples of measures, including n k-dimensional Hausdorff measure in R , Wiener measure and Brownian mo- tion, and Haar measure on topological groups. All these topics can be found in the references. c John K. Hunter, 2011 Contents Chapter 1. Measures 1 1.1. Sets 1 1.2. Topological spaces 2 1.3. Extended real numbers 2 1.4. Outer measures 3 1.5. σ-algebras 4 1.6. Measures 5 1.7. Sets of measure zero 6 Chapter 2. Lebesgue Measure on Rn 9 2.1. Lebesgue outer measure 10 2.2. Outer measure of rectangles 12 2.3. Carath´eodory measurability 14 2.4. Null sets and completeness 18 2.5. Translational invariance 19 2.6. Borel sets 20 2.7. Borel regularity 22 2.8. Linear transformations 27 2.9. Lebesgue-Stieltjes measures 30 Chapter 3. Measurable functions 33 3.1. Measurability 33 3.2. Real-valued functions 34 3.3.
    [Show full text]
  • Integral Geometry – Measure Theoretic Approach and Stochastic Applications
    Integral geometry – measure theoretic approach and stochastic applications Rolf Schneider Preface Integral geometry, as it is understood here, deals with the computation and application of geometric mean values with respect to invariant measures. In the following, I want to give an introduction to the integral geometry of polyconvex sets (i.e., finite unions of compact convex sets) in Euclidean spaces. The invariant or Haar measures that occur will therefore be those on the groups of translations, rotations, or rigid motions of Euclidean space, and on the affine Grassmannians of k-dimensional affine subspaces. How- ever, it is also in a different sense that invariant measures will play a central role, namely in the form of finitely additive functions on polyconvex sets. Such functions have been called additive functionals or valuations in the literature, and their motion invariant specializations, now called intrinsic volumes, have played an essential role in Hadwiger’s [2] and later work (e.g., [8]) on integral geometry. More recently, the importance of these functionals for integral geometry has been rediscovered by Rota [5] and Klain-Rota [4], who called them ‘measures’ and emphasized their role in certain parts of geometric probability. We will, more generally, deal with local versions of the intrinsic volumes, the curvature measures, and derive integral-geometric results for them. This is the third aspect of the measure theoretic approach mentioned in the title. A particular feature of this approach is the essential role that uniqueness results for invariant measures play in the proofs. As prerequisites, we assume some familiarity with basic facts from mea- sure and integration theory.
    [Show full text]
  • Invariant Measures on Locally Compact Groups
    INVARIANT MEASURES ON LOCALLY COMPACT GROUPS JENS GERLACH CHRISTENSEN Abstract. This is a survey about invariant integration on locally compact groups and its uses. The existence of a left invariant reg- ular Borel measure on locally compact Hausdorff groups is proved. It is also proved that this measure is unique in some sense. A few examples of interesting locally compact groups are given. Introduction An important property of the Lebesgue measure on Rn is that it is translation invariant. This means that translation of a measurable set does not change the value of its measure. Invariant measures are important tools in many areas of mathe- matics. For example the uncertainty principle related to Lie groups presented in [1] does not include any statements about an invariant measure, but the measure plays an important role in proving the the- orem. On a Lie group we can construct an invariant measure from the Lebesgue measure on the Lie algebra. As will be shown in this paper the group structure also gives rise to an invariant measure. That these two measures are essentially the same then follows from the uniqueness statement also to be found in the present paper. It is interesting to see how the Lebesgue measure on Rn can be generalised to groups. I would like to thank my supervisor Prof. Gestur Olafsson´ for his proofreading and guidance. 1. Locally compact groups In this section we introduce the notion of locally compact groups and regular measures. Definition 1.1. If G is a group and T is a topology on G such that (x, y) 7→ x−1 · y is a continuous map from G × G → G then G is called a topological group.
    [Show full text]
  • Haar Measure on Locally Compact Topological Groups
    Haar Measure on Locally Compact Topological Groups Abhishek Gupta February 2, 4, 2015 These are notes of my presentation delivered as a part of the course MTH 638: Abstract Harmonic Analysis at IIT Kanpur. Please send errors to [email protected]. The existence and uniqueness (i.e. upto scaling) of the Haar measure is proved. We follow the proof given by Weil and discussed in the book `Abstract Harmonic Analysis' by G.B. Folland. The modular function is also discussed briefly. By a locally compact group we shall mean a topological group which is locally compact and Hausdorff as a topological space. Note that in such a space the Urysohn lemma applies which we shall use, often without explicit comment. 1 Haar Measure: its Existence and Uniqueness Let G be a locally compact group. Cc(G) denotes the space of real valued compactly supported functions + on G and Cc (G) = ff 2 Cc(G): f ≥ 0; f 6= 0g. Recall that a Radon measure on a topological space X is a Borel measure which is finite on compact subsets (compact subsets of Hausdorff spaces are Borel measurable), outer regular on all Borel subsets and inner regular on all σ-finite Borel sets. Definition 1.1. A left(respectively right) Haar measure on a locally compact topological group G is a Radon measure µ such that µ(E) = µ(xE)(respectively µ(E) = µ(Ex)) for every Borel set E and 8x 2 G. Given a left Haar measure µ, one can immediately see thatµ ~ defined byµ ~(E) = µ(E−1) is a right Haar measure.
    [Show full text]
  • Haar Measure for Measure Groupoids1
    AMERICAN MATHEMATICAL SOCIETY Volume 242, August 1978 HAAR MEASUREFOR MEASURE GROUPOIDS1 BY PETER HAHN Abstract. It is proved that Mackey's measure groupoids possess an ana- logue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized. Introduction. In their pioneering investigation of operator algebras on Hubert space, Murray and von Neumann [12] obtained examples of non-type I factors using ergodic actions of a group on a measure space. Since then their method has been adapted and generalized several times to yield interesting new examples of von Neumann algebras; but despite recasting, the procedure has always seemed special and somewhat mystifying. The present work is the first of several papers treating Haar measure and convolution algebras of functions on George Mackey's measure groupoids [10]. This research permits us to interpret the construction made by Murray and von Neumann, as well as subsequent generalizations, in terms of these convolution algebras. Our unified treatment includes Krieger's construction of factors from nonfree actions of countable groups [7], Dixmier's examples of quasi-unitary algebras [2], and the regular representation of second count- able locally compact groups. They all arise from modular Hubert algebras of functions on some appropriate groupoid. Our approach seems natural because each step can easily be related to the special case of groups, which is widely known. Consider, for example, the case of a group action. Let g be a locally compact second countable group with Haar measure h; and suppose (5", /x) is a standard finite measure space on which g acts so that ¡x remains invariant.
    [Show full text]
  • Measures from Infinite-Dimensional Gauge Integration Cyril Levy
    Measures from infinite-dimensional gauge integration Cyril Levy To cite this version: Cyril Levy. Measures from infinite-dimensional gauge integration. 2019. hal-02308813 HAL Id: hal-02308813 https://hal.archives-ouvertes.fr/hal-02308813 Preprint submitted on 8 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Measures from infinite-dimensional gauge integration Cyril Levy IMT Toulouse – INUC Albi Email address: [email protected] January 2019 Abstract We construct and investigate an integration process for infinite products of compact metrizable spaces that generalizes the standard Henstock–Kurzweil gauge integral. The integral we define here relies on gauge functions that are valued in the set of divisions of the space. We show in particular that this integration theory provides a unified setting for the study of non- T absolute infinite-dimensional integrals such as the gauge integrals on R of Muldowney and the con- struction of several type of measures, such as the Lebesgue measure, the Gaussian measures on Hilbert spaces, the Wiener measure, or the Haar measure on the infinite dimensional torus. Furthermore, we characterize Lebesgue-integrable functions for those measures as measurable absolutely gauge inte- grable functions.
    [Show full text]
  • Tate's Thesis
    TATE'S THESIS BJORN POONEN These are an approximation of what was covered in lecture in 18.786 Number theory II at MIT during the first month or so of Spring 2015. This PDF file is divided into sections; the instructions for viewing the table of contents depend on which PDF viewer you are using. The syllabus for the semester was • Tate's thesis • Galois cohomology • An introduction to Galois representations (as much as we have time for) but these notes cover only Tate's thesis. We discuss both the number field and global function field cases, and also the subsequent reformulation by Deligne using L- and -factors. I thank Pavel Etingof, Sol Friedberg, Kenny Lau, and Jianing Li for some comments on these notes. February 3 1. The prototype: analytic continuation and functional equation of the Riemann zeta function Recall the Riemann zeta function, defined for Re s > 1 by Y X ζ(s) := (1 − p−s)−1 = n−s: prime p n≥1 Riemann gave two proofs of the following: Theorem 1.1 (Riemann 1860). (a) (Analytic continuation) The function ζ(s) extends to a meromorphic function on C, holomorphic except for a simple pole at s = 1. (b) (Functional equation) The completed zeta function s ξ(s) := π−s=2 Γ ζ(s) 2 defined below satisfies ξ(s) = ξ(1 − s). Today we will explain one of these proofs, as presented in [Dei05, Appendix A]. Hecke in 1918{1920 generalized Theorem 1.1 to Dedekind zeta functions ζK (s), and then quickly realized that his method applied also to Dirichlet L-functions and even more general 1 L-functions.
    [Show full text]
  • CHAPTER 3. P-ADIC INTEGRATION Contents
    CHAPTER 3. p-ADIC INTEGRATION Contents 1. Basics on p-adic fields1 2. p-adic integration5 3. Integration on K-analytic manifolds8 4. Igusa's theorem on the rationality of the zeta function 15 5. Weil's measure and the relationship with rational points over finite fields 19 References 22 The aim of this chapter is to set up the theory of p-adic integration to an extent which is sufficient for proving Igusa's theorem [Ig] on the rationality of the p-adic zeta function, and Weil's interpretation [We1] of the number of points of a variety over a finite field as a p-adic volume, in case this variety is defined over the ring of integers of a p- adic field. After recalling a few facts on p-adic fields, I will introduce p-adic integrals, both in the local setting and with respect to a global top differential form on a K- analytic manifold. I will explain Igusa's proof of the rationality of the zeta function using Hironaka's resolution of singularities in the K-analytic case, and the change of variables formula. Weil's theorem on counting points over finite fields via p-adic integration will essentially come as a byproduct; it will be used later in the course to compare the number of rational points of K-equivalent varieties. 1. Basics on p-adic fields We will first look at different approaches to constructing the p-adic integers Zp and the p-adic numbers Qp, as well as more general rings of integers in p-adic fields, and recall some of their basic properties.
    [Show full text]
  • Math 595: Geometric Measure Theory
    MATH 595: GEOMETRIC MEASURE THEORY FALL 2015 0. Introduction Geometric measure theory considers the structure of Borel sets and Borel measures in metric spaces. It lies at the border between differential geometry and topology, and services a variety of areas: partial differential equations and the calculus of variations, geometric function theory, number theory, etc. The emphasis is on sets with a fine, irregular structure which cannot be well described by the classical tools of geometric analysis. Mandelbrot introduced the term \fractal" to describe sets of this nature. Dynamical systems provide a rich source of examples: Julia sets for rational maps of one complex variable, limit sets of Kleinian groups, attractors of iterated function systems and nonlinear differential systems, and so on. In addition to its intrinsic interest, geometric measure theory has been a valuable tool for problems arising from real and complex analysis, harmonic analysis, PDE, and other fields. For instance, rectifiability criteria and metric curvature conditions played a key role in Tolsa's resolution of the longstanding Painlev´eproblem on removable sets for bounded analytic functions. Major topics within geometric measure theory which we will discuss include Hausdorff measure and dimension, density theorems, energy and capacity methods, almost sure di- mension distortion theorems, Sobolev spaces, tangent measures, and rectifiability. Rectifiable sets and measures provide a rich measure-theoretic generalization of smooth differential submanifolds and their volume measures. The theory of rectifiable sets can be viewed as an extension of differential geometry in which the basic machinery and tools of the subject (tangent spaces, differential operators, vector bundles) are replaced by approxi- mate, measure-theoretic analogs.
    [Show full text]
  • Weingarten Calculus and the Inthaar Package for Integrals Over Compact Matrix Groups
    Weingarten Calculus and the IntHaar Package for Integrals over Compact Matrix Groups Alejandro Ginory and Jongwon Kim Abstract In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals. 1 Introduction Integration over compact Lie groups with respect to the Haar measure is of great importance in a multitude of areas such as physics, statistics, and combinatorics to name a few. In [1], B. Collins establishes a method of symbolic integration over the unitary group which is later also developed for the orthogonal and the symplectic groups in [2],[3]. This method of integration is now known as Weingarten calculus and gives a combinatorial method for evaluating the integrals of polynomials with respect to the Haar measure. In this paper, we use Jack polynomials and twisted Gelfand pairs to give a unified presentation of Weingarten calculus. Using this presentation and combinatorial insights, we produce new optimization algorithms that greatly improve the efficiency and speed of computing these integrals. Finally, we present our implementation of the optimized Weingarten calculus for the unitary, orthogonal, and symplectic groups into a Maple package called IntHaar. Previously, the only package freely available online for Weingarten calculus dealt solely with the unitary group.
    [Show full text]
  • Toeplitz Operators on Locally Compact Abelian Groups David Gaebler Harvey Mudd College
    Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2004 Toeplitz Operators on Locally Compact Abelian Groups David Gaebler Harvey Mudd College Recommended Citation Gaebler, David, "Toeplitz Operators on Locally Compact Abelian Groups" (2004). HMC Senior Theses. 163. https://scholarship.claremont.edu/hmc_theses/163 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Toeplitz Operators on Locally Compact Abelian Groups David J. Gaebler Henry A. Krieger, Advisor Michael R. Raugh, Reader May, 2004 Department of Mathematics Abstract Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this cor- responds to forming a matrix out of the Fourier coefficients in a particu- lar way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators. Contents Abstract iii 1 The General Problem 1 2 The Unit Circle 3 2.1 Minimum Value of Qn ...................... 6 2.2 Szego’s¨ Theorem . 9 2.3 Connecting Szego¨ to Toeplitz . 14 2.4 Ratios to Roots . 15 2.5 Vitali and Weierstrass . 16 2.6 From Continuous Functions to Indicators . 17 2.7 Extending to Finite Measures . 19 3 The q-Torus 23 3.1 First Idea: Bijection from Zq to Z .
    [Show full text]