Constructive Function Theory in the Complex Plane Through Potential Theory and Geometric Function Theory (pp. 3-24)

摘要

This is a survey of some recent results concerning Bernstein-type inequalities for entirefunctions of the exponential tipe, harmonic majorants for classes of subharmonic functions,Phragmen-Lindel¨of function, the Krein description of the Cartwrite classes of entire functions withthe finite logarithmic integral, structure of the Martin boundary of the Denjoy domains, smoothnessproperties of the Green function, etc. We generalize the classical Bernstein theorem concerningthe constructive description of classes of functions uniformly continuous on the real line. Approximationof continuous bounded functions by entire functions of exponential type on an unboundedclosed proper subset of the real line or on an unbounded quasismooth (in the sense of Lavrentiev)curve in the complex plane is studied. We discuss Totik’s extension of the classical Bernsteintheorem on polynomial approximation of piecewise analytic functions on a closed interval. Theresults are achieved by the application of methods and techniques of modern geometric functiontheory and potential theory.