SELECTED PROBLEMS IN CLASSICAL FUNCTION THEORY

@inproceedings{Bnteau2014SELECTEDPI,
  title={SELECTED PROBLEMS IN CLASSICAL FUNCTION THEORY},
  author={Catherine B{\'e}n{\'e}teau and Dmitry Khavinson},
  year={2014}
}
We discuss several problems in classical complex analysis that might appeal to graduate students and young researchers. Among them are possible extensions to multiply connected domains of the Neuwirth-Newman theorem regarding analytic functions with positive boundary values, characterizing domains by properties of best approximations of z by analytic functions in various metrics, and sharpening the celebrated Putnam inequality in the context of Toeplitz operators on Bergman spaces and the… 
2 Citations
Three Problems in Operator Theory and Complex Analysis
OF THE DISSERTATION Three Problems in Operator Theory and Complex Analysis by Cheng Chu Doctor of Philosophy in Mathematics, Washington University in St. Louis, 2016. Professor John McCarthy, Chair
Isoperimetric inequalities for Bergman analytic content
The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic

References

SHOWING 1-10 OF 24 REFERENCES
Approximating z in Hardy and Bergman Norms
We consider the problem of …nding the best analytic approximation in Smirnov and Bergman norm to general monomials of the type znzm. We show that in the case of approximation to z in the annulus (and
On an extremal problem in the theory of rational approximation
A FREE BOUNDARY PROBLEM RELATED TO SINGLE-LAYER POTENTIALS
Let › be a bounded domain in R n with su-ciently regular boundary i , and let J be the operator which maps a function f on i to the restriction Jf to i of its single layer potential. In the present
Analytic Functions in Smirnov Classes Ep with Real Boundary Values
Smirnov domains with non-smooth boundaries do admit non-trivial functions of Smirnov class with real boundary values. We will show that the existance of functions in Smirnov classes with real
Putnam's theorem, Alexander's spectral area estimate, and VMO
In this paper we show that ] f f is a bounded analytic function defined on the unit disk such that at each point of the unit circle the cluster set o f f has area zero, then f has vanishing mean
Symmetry and uniform approximation by analytic functions
In this paper we treat the problem of finding all the domains in C for which the uniform distance from the function z to the space of analytic functions is equal precisely to (2 area/perimeter). We
Projections of Polynomial Hulls
The isoperimetric inequality and rational approximation
Etude d'une nouvelle approche de l'inegalite isoperimetrique par la formule de Stokes complexe et la theorie de l'approximation rationnelle
and M
  • Reguera, On a sharp estimate for Hankel operators and Putnam’s inequality, arXiv:1305.5193v1,
  • 2013
An inequality for the area of hyponormal spectra
...
...