Harmonic Analysis and Hypergroups

Under the guidance and inspiration of Dr. Ajit Iqbal Singh, an International Conference on Harmonie Analysis took place at the Uni versity of Delhi, India, from December 18 to 22, 1995. Twenty-one dis tinguished mathematicians from around the world, as weIl as many from India, participated in this successful and stimulating conference. An underlying theme of the conference was hypergroups, the the ory of wh ich has developed and been found useful in fields as diverse as special functions, differential equations, probability theory, representa tion theory, measure theory, Hopf algebras and quantum groups. Some other areas of emphasis that emerged were harmonie analysis of analytic functions, ergo die theory and wavelets. This book includes most of the proceedings of this conference. I chaired the Editorial Board for this publication; the other members were J. M. Anderson (University College London), G. L. Litvinov (Centre for Optimization and Mathematical Modeling, Institute for New Technolo gies, Moscow), Mrs. A. I. Singh (University ofDelhi, India), V. S. Sunder (Institute of Mathematical Sciences, C.LT., Madras, India), and N. J. Wildberger (University of New South Wales, Australia). I appreciate all the help provided by these editors as weIl as the help and cooperation of Our authors and referees of their papers. I especially appreciate techni cial assistance and advice from Alan L. Schwartz (University of Missouri - St. Louis, USA) and Martin E. Walter (University of Colorado, USA). Finally, I thank Our editor, Ann Kostant, for her help and encouragement during this project.

Includes supplementary material: sn.pub/extras

Contenu
Contents Preface 1 Introduction 1 The Set N of Natural Numbers 2 The Set Q of Rational Numbers 3 The Set R of Real Numbers 4 The Completeness Axiom 5 The Symbols + and - 6 A Development of R 2 Sequences 7 Limits of Sequences 8 A Discussion about Proofs 9 Limit Theorems for Sequences 10 Monotone Sequences and Cauchy Sequences 11 Subsequences 12 limsup's and liminf's 13 Some Topological Concepts in Metric Spaces 14 Series 15 Alternating Series and Integral Tests 16 Decimal Expansions of Real Numbers 3 Continuity 17 Continuous Functions 18 Properties of Continuous Functions 19 Uniform Continuity 20 Limits of Functions 21 More on Metric Spaces: Continuity 22 More on Metric Spaces: Connectedness 4 Sequences and Series of Functions 23 Power Series 24 Uniform Convergence 25 More on Uniform Convergence 26 Differentiation and Integration of Power Series 27 Weierstrass's Approximation Theorem 5 Differentiation 28 Basic Properties of the Derivative 29 The Mean Value Theorem 30 L'Hospital's Rule 31 Taylor's Theorem 6 Integr ation 32 The Riemann Integral 33 Properties of the Riemann Integral 34 Fundamental Theorem of Calculus 35 Riemann-Stieltjes Integrals 36 Improper Integrals 37 A Discussion of Exponents and Logarithms Appendix on Set Notation Selected Hints and Answers References Index