TABLE OF CONTENTS

1. Introduction 1

2. Notation and Preliminaries 7

3. The

Lp

Schwartz Spaces 13

4. The Principal Series 15

5. The Discrete Series 19

6. Leading Exponents and Distributions 23

7. Relationships between Principal and Discrete Series Matrix Coefficients 29

8. The Trombi-Varadarajan Estimates for SL(2, R) 33

9. The Fourier Transform on

CP(G)

37

10. The Plancherel Inversion Formula 46

11. The Decomposition of

CP(G)

49

12. Asymptotic Approximation of Matrix Coefficients 53

13. Growth of Asymptotic Coefficients for the Principal Series 62

14. Calculation of Asymptotic Coefficents for the Discrete Series 70

15. The Inverse Transform 76

16. The Isomorphism Theorem: Non-Integral Case 88

17. The Campoli Functions 92

18. The Isomorphism Theorem: General Case 97

19. The Zero-Schwartz Space (with Henrik Schlichtkrull) 99

List of Notation 107

References 109

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