Analysis IV

1. Front Matter

   Pages I-XI

   PDF

2. XI – Integration and Fourier Transform

   1. Front Matter

      Pages 1-4

      PDF

   2. The Upper Integral of a Positive Function

      • Roger Godement

      Pages 5-19

   3. L ^p Spaces

      • Roger Godement

      Pages 20-36

   4. Measurable Sets and Functions

      • Roger Godement

      Pages 37-52

   5. Lebesgue-Fubini’s Way

      • Roger Godement

      Pages 53-102

   6. The Lebesgue-Nikodym Theorem

      • Roger Godement

      Pages 103-117

   7. Spectral Decomposition on a Hilbert Space

      • Roger Godement

      Pages 118-172

   8. The Commutative Fourier Transform

      • Roger Godement

      Pages 173-203

   9. Unitary Representations of Locally Compact Groups

      • Roger Godement

      Pages 204-259

3. XII –The Garden of Modular Delights or The Opium of Mathematicians

   1. Front Matter

      Pages 261-261

      PDF

   2. Infinite Series and Products in Number Theory

      • Roger Godement

      Pages 261-280

   3. The series \(\sum 1/ {\rm{cos}} \ \pi n z\; {\rm{and}}\; \sum {\rm{exp}} \left(\pi in^2z\right)\)

      • Roger Godement

      Pages 281-297

   4. The Dirichlet Series \(L(s; \mathcal{X})\)

      • Roger Godement

      Pages 298-315

   5. Elliptic Functions

      • Roger Godement

      Pages 316-348

   6. SL ₂(ℝ) as a Locally Compact Group

      • Roger Godement

      Pages 349-375

   7. Modular Functions: The Classical Theory

      • Roger Godement

      Pages 376-393

   8. Fuchsian Groups

      • Roger Godement

      Pages 394-440

   9. Hecke Theory

      • Roger Godement

      Pages 441-462

   10. SL ₂(ℝ) as a Lie Group

       • Roger Godement

       Pages 463-509

Analysis Volume IV introduces the reader to functional analysis (integration, Hilbert spaces, harmonic analysis in group theory) and to the methods of the theory of modular functions (theta and L series, elliptic functions, use of the Lie algebra of SL2). As in volumes I to III, the inimitable style of the author is recognizable here too, not only because of his refusal to write in the compact style used nowadays in many textbooks. The first part (Integration), a wise combination of mathematics said to be `modern' and `classical', is universally useful whereas the second part leads the reader towards a very active and specialized field of research, with possibly broad generalizations.

• compact level spaces
• elliptic functions
• integrable functions
• modular functions
• spaces
• spectral theory
• zeta functions

“The author includes many interesting historical comments of great value for those interested in the history of mathematics. The author includes sometimes examples and exercises which help the reader to understand the theory presented.” (Dumitrŭ Popa, zbMATH 1331.28001, 2016)

• Paris, France

  Roger Godement

Roger Godement (October 1, 1921 - July 21, 2016) is known for his work in functional analysis, and also his expository books. He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. He also wrote texts on Lie groups, abstract algebra and mathematical analysis.

Policies and ethics