This is a three year program in classical analysis, in particular in potential theory with applications in approximation theory and orthogonal polynomials. A common unifying theme is the behavior of Christoffel functions and reproducing kernels and their various applications. Another unifying theme is the so called polynomial inverse image method. The applications include fine zero behavior of orthogonal polynomials (with direct translation to eigenvalues of Jacobi matrices), universality results in random matrix theory/statistical physics and various polynomial inequalities. Other questions are related to non-classical orthogonal polynomials (with respect to doubling weights) and approximation by homogeneous polynomials and by their level sets. In several of these problems the polynomial inverse image method - that has already produced sharp and significant results in the past - will play a crucial role, and, in return, the tools to be developed will help us in better understanding this powerful technique. At the heart of the research will be the systematic usage of potential theory and classical harmonic analysis in the relatively distant areas of approximation theory and orthogonal polynomials.
The proposed study of various properties of orthogonal polynomials is aimed, by introducing there new tools, to advance a very classical field in mathematics (going back to over 200 years) which has multitudes of connections and applications. The results are relevant to other branches of mathematics, physics and engineering, as well. The research will stimulate interest in students and enhance research environment for them. Graduate and PhD students will have the opportunity to learn the fundamentals and powerful techniques of different disciplines, as well as their interrelations. Some of the results and methods will be integrated into related courses, which in turn may advance the professional development of K-12 science and mathematics teachers. Special emphasis will be made to outreach general public. This will be achieved by publishing educational articles, thereby offering understanding and appreciation for science. These educational materials, just as the findings of the research, will be widely distributed. Lectures will be held at different levels from undergraduate societies to professionals meetings.
This was a project in mathematics, in particular in mathematical analysis, which is a classical area of mathematics based on calculus. The project dealt with basic research, its primary aim was to understand and advance the theory which later may have applications outside mathematical analysis. The main focus was on problems lying in approximation theory and the theory of orthogonal polynomials. Approximation theory is the area in mathematics that deals with the problem how one can substitute complicated, sometimes indescribably objects by simpler ones. Polynomials and orthogonal polynomials are such simpler objects, they are easy to generate, and they are capable of displaying any behavior that other system exhibit. In the project several problems have been solved in connection with polynomials and orthogonal polynomials. In particular, a long standing open problem on polynomial approximation in higher dimension has been completely settled (the solution appears in a small book of about 110 pages). The solution came by proving a result from the middle 1980's in complete generality (the result itself was for very special polytopes, and the conjecture had been that the same statement is true for all convex polyhedra, and that is precisely what has been done). The project gave the opportunity to two graduate students to get acquainted with some of the techniques used in the theory and to do some basic research themselves.
