Final Report Summary - POTENTIALTHEORY (Potential theoretic methods in approximation and orthogonal polynomials)
Potential theory has its roots in physics, but has emerged as a separate mathematical discipline in the first half of the last century. The present proposal has been basic research in theoretical mathematics, it is aimed at developing potential theoretical tools that can be used in other disciplines, namely in approximation theory and in the theory of orthogonal polynomials. Approximation theory deals with the problem of replacing complicated objects with simpler ones that can be handled more easily and can be used in practice, while orthogonal polynomials give such tools. These are disciplines lying rather far from potential theory, but their interaction with the latter discipline has been particularly fruitful in the last 2 decades. In some of the questions even the correct formulation of the problem requires potential theoretical concepts such as equilibrium and capacity.
The PI has been working within the project with two PhD students and one postdoc on several problems in this interaction. His students have written 9 papers (6 have appeared) and both of them got their PhD in the framework of the project.
With the support of the project a successful conference was organized in 2012 in the host town Szeged with the title “Workshop on Potential Theory and its Applications” (http://www.math.u-szeged.hu/wspota2012/). 40 experts from potential theory, approximation theory and orthogonal polynomials from 16 countries took part, giving about 30 lectures.
Within the project one research monograph has been published in the Memoirs series of the American Mathematical Society – it solves (together with another paper written during the project) a long standing open problem about higher dimensional approximation by the simplest possible objects: polynomials. In connection with various problems set forth in the original project, 35 research papers have been, and four others will be published.
During the project the PI was invited to give 13 plenary addresses and 5 invited talks on international conferences, and his student and postdoc gave 15 contributed talks. As wider dissemination of the research and to make the mathematics discipline more attractive, the PI gave 7, and his students gave 8 general audience presentations on high school and university student events and society meetings.
The research in the project was in various directions of mathematical analysis that required new tools and ideas. An underlying unifying theme was the systematic use of potential theory. The major directions were polynomial approximation in several variables (approximation on polytopes), orthogonal polynomials (universality questions and zero spacing), Christoffel functions (asymptotic formulae on Jordan curves and arcs and for doubling measures), inequalities for polynomials and rational functions (Markov and Bernstein type inequalities on Jordan curves and arcs, best constants), the size of polynomials (Chebyshev polynomials, Widom’s conjecture) and locations of their zeros (discrepancy results, asymptotic Gauss-Lucas theorem).
The PI has been working within the project with two PhD students and one postdoc on several problems in this interaction. His students have written 9 papers (6 have appeared) and both of them got their PhD in the framework of the project.
With the support of the project a successful conference was organized in 2012 in the host town Szeged with the title “Workshop on Potential Theory and its Applications” (http://www.math.u-szeged.hu/wspota2012/). 40 experts from potential theory, approximation theory and orthogonal polynomials from 16 countries took part, giving about 30 lectures.
Within the project one research monograph has been published in the Memoirs series of the American Mathematical Society – it solves (together with another paper written during the project) a long standing open problem about higher dimensional approximation by the simplest possible objects: polynomials. In connection with various problems set forth in the original project, 35 research papers have been, and four others will be published.
During the project the PI was invited to give 13 plenary addresses and 5 invited talks on international conferences, and his student and postdoc gave 15 contributed talks. As wider dissemination of the research and to make the mathematics discipline more attractive, the PI gave 7, and his students gave 8 general audience presentations on high school and university student events and society meetings.
The research in the project was in various directions of mathematical analysis that required new tools and ideas. An underlying unifying theme was the systematic use of potential theory. The major directions were polynomial approximation in several variables (approximation on polytopes), orthogonal polynomials (universality questions and zero spacing), Christoffel functions (asymptotic formulae on Jordan curves and arcs and for doubling measures), inequalities for polynomials and rational functions (Markov and Bernstein type inequalities on Jordan curves and arcs, best constants), the size of polynomials (Chebyshev polynomials, Widom’s conjecture) and locations of their zeros (discrepancy results, asymptotic Gauss-Lucas theorem).