The main goal of this proposal is to investigate a series of fundamental problems in constructive function theory which constitute common ground of the analysis and applied mathematics. Our approach borrows ideas and techniques from many fields of theoretical mathematics, such as real and complex analysis, topology, and Fourier analysis. Recently, Carleson, Totik, and the PI in a series of papers have found a new approach to connect the continuous properties of the Green function and the metric properties of the boundary of a domain where the Green function is defined. We believe that this approach can give a decisive impulse to investigation of a number of long-standing open problems in constructive functio theory. A major component of this proposal is to study a new representation of basic notions of potential theory (logarithmic capacity, the Green function, and equilibrium measure) in terms of a conformal mapping of the exterior of the unit interval onto the exterior of the unit disk with finite or infinite number of radial slits, presented in the recent work of the PI. We analyze the geometry of Cantor-type sets and propose to find a new proof to and significant extension of the results by Totik and Carleson on sets possessing the Hoelder continuous Green function. The second part of our proposal concerns Markov- and Remez-type inequalities for polynomials on subsets of the real line. We propose to construct a general $2$-dimensional theory of Remez-type inequalities and illustrate their power by giving a number of applications. The last part is devoted to study of well-known open problems in polynomial approximation in the complex plane which have a large number of applications in both pure and applied mathematics. We hope to find a complete solution of the Meinardus-Varga problem on structure of an entire function with the geometric convergence on the positive real axis of reciprocals of polynomials to the reciprocal of the function. We intend to employ a new concept of Faber-type polynomials. Our prior reseach indicates that there exists a connection between th Nikolskii-Timan-Dzjadyk approximation theorem and the concept of uniformly perfect sets introduced by Beardon and Pommerenke. We propose to investigate the details of this connection.
The major component of the broader impact of our proposal is to create a new link between potential theory, geometric function theory and constructive function theory. Another component concerns the training of graduate and undergraduate students. Indeed, the problems addressed in this proposal are stated in such a way that not only are they clear to the graduate students but they are accessible to undergraduates as well. On the other hand, the answers to many of those problems are quite counterintuitive. This stimulates the interest of students to the subject and mathematics in general.
