Benedicks / Jones / Smirnov Perspectives in Analysis

Essays in Honor of Lennart Carleson's 75th Birthday
1. Auflage 2006
ISBN: 978-3-540-30434-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 27, 378 Seiten, eBook

Reihe: Mathematical Physics Studies

The Conference “Perspectives in Analysis” was held during May 26–28, 2003 at the Royal Institute of Technology in Stockholm, Sweden. The purpose of the conference was to consider the future of analysis along with its relations to other areas of mathematics and physics, and to celebrate the seventy-?fth birthday of Lennart Carleson. The scienti?c theme was one with which the name of Lennart Carleson has been associated for over ?fty years. His modus operandi has long been to carry out a twofold approach to the selection of research problems. First one should look for promising new areas of ana- sis, especially those having close contact with physically oriented problems of geometric character. The second step is to select a core set of problems that require new techniques for their resolutions. After making a central contri- tion, Lennart would usually move on to a new area, though he might return to the topic of his previous work if new techniques were developed that could break old mathematical log jams. Lennart’s operating approach is based on fundamental realities of modern mathematics as well as his own inner c- victions. Here we ?rst refer to an empirical fact of mathematical research: All topics have a ?nite half-life, with ?fteen years being an upper bound for most areas. After that time it is usually a good idea to move on to so- thing new.

Zielgruppe

Research

Autoren/Hrsg.

Benedicks, Michael

Jones, Peter

Smirnov, Stanislav

Inhaltsverzeichnis

The Rosetta Stone of L-functions.- New Encounters in Combinatorial Number Theory: From the Kakeya Problem to Cryptography.- Perspectives and Challenges to Harmonic Analysis and Geometry in High Dimensions: Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data.- Open Questions on the Mumford-Shah Functional.- Multi-scale Modeling.- Mass in Quantum Yang-Mills Theory (Comment on a Clay Millennium Problem).- On Scaling Properties of Harmonic Measure.- The Heritage of Fourier.- The Quantum-Mechanical Many-Body Problem: The Bose Gas.- Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle.- Heat Measures and Unitarizing Measures for Berezinian Representations on the Space of Univalent Functions in the Unit Disk.- On Local and Global Existence and Uniqueness of Solutions of the 3D Navier—Stokes System on ?3.- Analysis on Lie Groups: An Overview of Some Recent Developments and Future Prospects.- Encounters with Science: Dialogues in Five Parts.

Leseproben

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 [e-mail]
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.