Selected Works of Donald L. Burkholder

Selected Works of Donald L. Burkholder
Author :
Publisher : Springer Science & Business Media
Total Pages : 715
Release :
ISBN-10 : 9781441972453
ISBN-13 : 1441972455
Rating : 4/5 (53 Downloads)

Synopsis Selected Works of Donald L. Burkholder by : Burgess Davis

This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.

Brownian Motion

Brownian Motion
Author :
Publisher : Walter de Gruyter GmbH & Co KG
Total Pages : 533
Release :
ISBN-10 : 9783110741278
ISBN-13 : 311074127X
Rating : 4/5 (78 Downloads)

Synopsis Brownian Motion by : René L. Schilling

Stochastic processes occur everywhere in the sciences, economics and engineering, and they need to be understood by (applied) mathematicians, engineers and scientists alike. This book gives a gentle introduction to Brownian motion and stochastic processes, in general. Brownian motion plays a special role, since it shaped the whole subject, displays most random phenomena while being still easy to treat, and is used in many real-life models. Im this new edition, much material is added, and there are new chapters on ''Wiener Chaos and Iterated Itô Integrals'' and ''Brownian Local Times''.

Selected Papers of Alberto P. Calderon with Commentary

Selected Papers of Alberto P. Calderon with Commentary
Author :
Publisher : American Mathematical Soc.
Total Pages : 686
Release :
ISBN-10 : 0821842978
ISBN-13 : 9780821842973
Rating : 4/5 (78 Downloads)

Synopsis Selected Papers of Alberto P. Calderon with Commentary by : Alberto P. Calderón

Alberto Calderon was one of the leading mathematicians of the twentieth century. His fundamental, pioneering work reshaped the landscape of mathematical analysis. This volume presents a wide selection from some of Calderon's most influential papers. They range from singular integrals to partial differential equations, from interpolation theory to Cauchy integrals on Lipschitz curves, from inverse problems to ergodic theory. The depth, originality, and historical impact of these works are vividly illustrated by the accompanying commentaries by some of today's leading figures in analysis. In addition, two biographical chapters preface the volume. They discuss Alberto Calderon's early life and his mathematical career.

Harmonic Analysis

Harmonic Analysis
Author :
Publisher : American Mathematical Soc.
Total Pages : 779
Release :
ISBN-10 : 9781470411022
ISBN-13 : 1470411024
Rating : 4/5 (22 Downloads)

Synopsis Harmonic Analysis by : Barry Simon

A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 3 returns to the themes of Part 1 by discussing pointwise limits (going beyond the usual focus on the Hardy-Littlewood maximal function by including ergodic theorems and martingale convergence), harmonic functions and potential theory, frames and wavelets, spaces (including bounded mean oscillation (BMO)) and, in the final chapter, lots of inequalities, including Sobolev spaces, Calderon-Zygmund estimates, and hypercontractive semigroups.

Martingales in Banach Spaces

Martingales in Banach Spaces
Author :
Publisher : Cambridge University Press
Total Pages : 591
Release :
ISBN-10 : 9781107137240
ISBN-13 : 1107137241
Rating : 4/5 (40 Downloads)

Synopsis Martingales in Banach Spaces by : Gilles Pisier

This book focuses on applications of martingales to the geometry of Banach spaces, and is accessible to graduate students.

Selected Works of Donald L. Burkholder

Selected Works of Donald L. Burkholder
Author :
Publisher : Springer
Total Pages : 729
Release :
ISBN-10 : 1493940694
ISBN-13 : 9781493940691
Rating : 4/5 (94 Downloads)

Synopsis Selected Works of Donald L. Burkholder by : Burgess Davis

This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.

Proceedings of the Norbert Wiener Centenary Congress, 1994

Proceedings of the Norbert Wiener Centenary Congress, 1994
Author :
Publisher : American Mathematical Soc.
Total Pages : 622
Release :
ISBN-10 : 0821867571
ISBN-13 : 9780821867570
Rating : 4/5 (71 Downloads)

Synopsis Proceedings of the Norbert Wiener Centenary Congress, 1994 by : Vidyadhar Mandrekar

A mathematician on par with the greatest in the century, Norbert Wiener was a universal thinker of colossal proportions. This book contains the proceedings of the Norbert Wiener Centenary Congress held at Michigan State University on November 27-December 2, 1994. The aim of the Congress was to reveal the depth and strong coherence of thought that runs through Wiener's legacy, and to exhibit its continuation in on-going research. This volume brings together the great minds who have furthered Wiener's ideas in physics, stochastics, harmonic analysis, philosophy, prosthesis and cybernetics. The presentations coherently lay out the developments of the subjects from their inception. This volume provides an excellent pathway for new investigators who may wish to pursue these developments by following the footsteps of world experts. There is no other book available in which experts in the various fields in which Wiener worked have presented his thoughts and contributions insuch a coherent and lucid manner.

Schrödinger Operators, Como 1984

Schrödinger Operators, Como 1984
Author :
Publisher : Springer
Total Pages : 282
Release :
ISBN-10 : 9783540397069
ISBN-13 : 354039706X
Rating : 4/5 (69 Downloads)

Synopsis Schrödinger Operators, Como 1984 by : Sandro Graffi