Stéphane Jaffard

from Wikipedia, the free encyclopedia

Stéphane Jaffard (born May 23, 1962 in Boulogne-Billancourt ) is a French mathematician who deals with harmonic analysis and fractals . He is a professor at the University of Paris XII (Marne-la-Vallée).

Life

Jaffard studied from 1981 to 1984 at the École Polytechnique , where he received his doctorate under Yves Meyer (Construction et propriétés des bases d'ondelettes, remarques sur la controlabilité exacte). In 1989/90 he was at the Institute for Advanced Study and from 1990 to 1992 at the École des Ponts ParisTech , where he was then director of CERMA (Center d'enseignement et de Recherche en Mathématiques Appliquée) until 1994. In 1992 he completed his habilitation at the University of Paris IX (Dauphine). In 1995 he became a professor at the University of Paris-East. In 1996 he became PEDR (Prime d'Encadrement Doctoral et de Recherche), in 1997 he became professor 1st class and in 2007 he became a professor of the class exceptionelle.

From 2000 to 2005 he was a junior member of the Institut de France. He was visiting scholar in Montreal, the Isaac Newton Institute , the University of British Columbia, the University of Vienna, Purdue University and the University of California, Riverside.

plant

He deals with wavelets and multifractals (multifractal analysis) with applications, for example in image processing. In 1991 he introduced Wilson bases with Ingrid Daubechies and Jean-Lin Journé . Its outstanding properties in the time and frequency analysis of signals were used in one of the algorithms that led to the direct discovery of gravitational waves in 2015 .

In 1996 he investigated and also determined the Hölder regularity (point-by-point investigation of the regularity) of Riemann's example of a nowhere differentiable function that fluctuates strongly from point to point in an irregular manner, depending on the properties of the Diophantine approximation of the point, and even discontinuously at every point is. He later showed that this is not the exception, but in some ways the generic case and is also found in many stochastic processes such as most of the Lévy processes.They provide examples of multifractals. Since the Hölder exponent can only be applied to locally restricted functions and this is not the case with many signal functions, p-exponents are investigated instead (according to Alberto Calderón and Antoni Zygmund ) and Jaffard was able to characterize the p-exponent with Clothilde Melot with wavelets give. With Martin Bruno he determined the p-exponents of the Brjuno function of Jean-Christophe Yoccoz from the theory of dynamic systems (which is nowhere locally restricted).

The local behavior of the Hölder exponent was related to their fractal behavior (exponent of the scaling function) by Uriel Frisch and Giorgio Parisi in 1985 . The non-linearity of the scaling function was an indication of different Hölder exponents in the signal. They also introduced the multifractal spectrum (Frisch-Parisi formula), which relates the fractal dimension of sets, on which the function has a certain Hölder exponent, to the scaling function. That was the beginning of the multifractal analysis that Jaffard expanded. He was able to show that the multifractal spectrum can often be described by simple functions, even if the behavior of the Hölder exponents was very complicated. To this end, he also introduced new wavelet transformation methods (wavelet leader method) and applied them, for example, to the theory of turbulence and the analysis of images by Vincent van Gogh (assignment to different creative periods, differentiation of forgeries, with Patrice Abry, Herwig Wendt).

Fonts (selection)

  • with Yves Meyer, R. Ryan: Wavelets. Tools for science and technology, SIAM 2001
  • with Y. Meyer: Wavelet Methods for Pointwise Regularity and Local Oscillation sof Functions, Memoirs of the AMS 123, 1996
  • with Alain Damlamian (Ed.): Wavelet methods in mathematical analysis and engineering, World Scientific 2010
  • The spectrum of singularities of Riemann's function, Revista Mathematica Iberoamericana, Volume 12, 1996, pp. 441-460
  • with A. Arneodo, E. Bacry, J.-F. Muzy: Oscillating singularities on Cantor sets: a grand canonical multifractal formalism, Journal of Statistical Physics, Volume 87, 1997, pp. 179-209
  • Multifractal formalism for functions, 2 parts, SIAM J. Math. Analysis, Volume 28, 1997, pp. 944-970, 971-998
  • The multifractal nature of Lévy processes, Probability Theory and Related Fields, Volume 114, 1999, pp. 207-227
  • with C. Melot: Wavelet analysis of fractal boundaries, Part 1: Local regularity, Communications in Mathematical Physics, Volume 258, 2005, pp. 513-539, Part 2: Multifractal formalism, pp. 541-565
  • with J.-M. Aubry: Random wavelet series, Comm. Math. Phys., Vol. 227, 2002, pp. 483-514
  • with A. Fraysse: How smooth is almost every function in a Sobolev space?, Revista Matematica Iberoamericana, Volume 22, 2006, pp. 663-682
  • with Eric Chassande-Mottin, Yves Meyer: Des ondelettes pour détecter les ondes gravitationnelles ,. Gazette des Mathématiciens, April 2016
  • with Benoit Mandelbrot: Peano-Polya motion, when time is intrinsic or binomial (uniform or multifractal), The Mathematical Intelligencer, 1997, issue 4
  • with Yves Meyer, O. Rioul: L'analyse par ondelettes, Pour la Science, September 1987
  • with P. Abry, H. Wendt: When Van Gogh meets Mandelbrot: Multifractal Classification of Painting's Texture, Signal Processing, Volume 93, 2013, pp. 554-572

Web links

References and comments

  1. ↑ Daubechies, Jaffard, Journé, A Simple Wilson Orthonormal Basis with Exponential Decay, SIAM J. Math. Analysis, Vol. 22, 1991, pp. 554-573
  2. ↑ Not with the Brownian motion or the Weierstrass function , there the Hölder exponent is the same everywhere and with the Brownian motion . Godfrey Harold Hardy suspected the irregularity of Bernhard Riemann's nowhere indifferentiable function in 1916, who determined the Hölder regularity of the Weierstrass function.
  3. ↑ With a linear scaling function the Hölder exponent is constant, for example with Brownian motion.
  4. ↑ In retrospect, this began with the investigation of homogeneous turbulence and its scaling behavior by Andrei Kolmogorow in the 1940s.
personal data
SURNAME Jaffard, Stéphane
BRIEF DESCRIPTION French mathematician
DATE OF BIRTH May 23, 1962
PLACE OF BIRTH Boulogne-Billancourt