Contructive Function Theory on Subsets of the Real Line Through Potential Theory and Geometric Function Theory

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ABSTRACT:The main goal of this proposal is to investigate a series of fundamentalproblems in constructive function theory which constitute common groundof the analysis and applied mathematics. Our approach borrows ideas andtechniques from many fields of theoretical mathematics, such as real andcomplex analysis, topology, and Fourier analysis. Recently, Carleson,Totik, and the PI in a series of papers have found a new approach toconnect the continuous properties of the Green function and the metricproperties of the boundary of a domain where the Green function isdefined. We believe that this approach can give a decisive impulse toinvestigation of a number of long-standing open problems in constructivefunctio theory. A major component of this proposal is to study a newrepresentation of basic notions of potential theory (logarithmic capacity, the Green function, andequilibrium measure) in terms of a conformal mapping of the exterior ofthe unit interval onto the exterior of the unit disk with finite orinfinite 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 newproof to and significant extension of the results by Totik and Carleson onsets possessing the Hoelder continuous Green function. The second part ofour proposal concerns Markov- and Remez-type inequalities for polynomialson subsets of the real line. We propose to construct a general$2$-dimensional theory of Remez-type inequalities and illustrate theirpower by giving a number of applications. The last part is devoted to study of well-knownopen problems in polynomial approximation in the complex plane which havea large number of applications in both pure and applied mathematics. Wehope to find a complete solution of the Meinardus-Varga problem onstructure of an entire function with the geometric convergence on thepositive real axis of reciprocals of polynomials to the reciprocal of thefunction. We intend to employ a new concept of Faber-type polynomials. Ourprior reseach indicates that there exists a connection between thNikolskii-Timan-Dzjadyk approximation theorem and the concept of uniformlyperfect sets introduced by Beardon and Pommerenke. We propose toinvestigate the details of this connection. The major component of the broader impact of our proposal is to create anew link between potential theory, geometric function theory andconstructive function theory. Another component concerns the trainingof graduate and undergraduate students. Indeed, the problems addressed inthis proposal are stated in such a way that not only are they clear to thegraduate students but they are accessible to undergraduates as well. Onthe other hand, the answers to many of those problems are quitecounterintuitive. This stimulates the interest of students to the subjectand mathematics in general.