Perspectives in Analysis (eBook)

Preface 7
Contents 11
List of Contributors 13
The Rosetta Stone of L- functions 15
New Encounters in Combinatorial Number Theory: From the Kakeya Problem to Cryptography 31
Perspectives and Challenges to Harmonic Analysis and Geometry in High Dimensions: Geometric Di . usions as a Tool for Harmonic Analysis and Structure De . nition of Data 41
Open Questions on the Mumford–Shah Functional 51
Multi-scale Modeling 65
Mass in Quantum Yang–Mills Theory ( Comment on a Clay Millennium Problem) 77
On Scaling Properties of Harmonic Measure 87
The Heritage of Fourier 97
The Quantum-Mechanical Many-Body Problem: The Bose Gas 111
Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle 199
Heat Measures and Unitarizing Measures for Berezinian Representations on the Space of Univalent Functions in the Unit Disk 267
On Local and Global Existence and Uniqueness of Solutions of the 3D Navier – Stokes System on R3 283
Analysis on Lie Groups: An Overview of Some Recent Developments and Future Prospects 297
Encounters with Science: Dialogues in Five Parts 309

The Heritage of Fourier ( p. 83)

Jean-Pierre Kahane

D´epartement de math´ematiques, Universit´e Paris-Sud Orsay, Bˆat. 425, F-91405 Orsay Cedex, France jean-pierre.kahane@math.u-psud.fr
1 Purpose of the Article
The heritage of Fourier is many-sided. First of all Fourier is a physicist and a mathematician. The name Fourier is familiar to mathematicians, physicists, engineers and scientists in general. The Fourier equation, meaning the heat equation, Fourier series, Fourier coe fficients, Fourier integrals, Fourier transforms, Fourier analysis, Fast Fourier Transforms, are everyday terms. The Analytical Theory of Heat is recognized as a landmark in science.

But Fourier is known also as an Egyptologist. He wrote an extensive introduction to the series of books entitled "Description de l’Egypte". He was in Egypt when the Rosetta stone was discovered, and Jean-Fran¸cois Champollion, who deciphered the hieroglyphs, was introduced in the subject by Fourier.

He was also an administrator and a politician. He took part in the French Revolution (Arago said that he was a pure product of the French Revolution, because he was supposed first to become a priest), he followed Bonaparte and Monge in Egypt as "secr´etaire perp´etuel de l’Institut d’Egypte", then Bonaparte elected him as prefect in Grenoble where he led a very important action in health and education, and he became a member of both Acad´emie des sciences and Acad´emie Fran¸caise when he settled back in Paris after the fall of Napoleon.

He was elected as Secr´etaire perp´etuel de l’Acad´emie des sciences and played a role for the recognition of statistics in France. His scientific work does not consist only in the analytical theory of heat and the tools that he created for this theory. He was interested in algebraic equations and his work on the localization of the roots is the transition from Descartes to Sturm, unfortunately he neglected Galois. He himself was neglected for his work on inequalities, what he called "Analyse ind´etermin´ee".

Darboux considered that he gave the subject an exaggerated importance and did not publish the papers on this question in his edition of the scientific works of Fourier. Had they been published, linear programming and convex analysis would be included in the heritage of Fourier.

Fourier was a learned man and a philosopher in the sense of the eighteenth century. In a way he is a late representative of the Age of Enlightenment. On the other hand he is the main reference for Auguste Comte, a starting point for the French "positivism" of the nineteenth century. I shall concentrate on a narrow but important part of his scienti.c heritage, namely the expansion of a function into a trigonometric series and the formulas for computing the coefficients. It is a way to enter the way of thinking of Fourier and its relation to physics and natural philosophy, as well as to explore the purely mathematical continuation of his work.