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The Schwarz function and its generalization to higher dimensions. (English) Zbl 0784.30036
The University of Arkansas Lecture Notes in the Mathematical Sciences. New York: John Wiley & Sons Ltd. xiv, 108 p. (1992).
The so-called Schwarz function comes from the following result. Proposition: Let \(\Gamma\) be a nonsingular analytic Jordan arc in \({\mathbf R}^ 2\cong{\mathbf C}\). Then there are a neighborhood \(\Omega\) of \(\Gamma\) and a unique holomorphic function \(S\) on \(\Omega\) such that \(S(z)=\bar z\) for \(z\in \Gamma\). Shades of Plemelj! Clearly, this result is closely related to (anti-holomorphic) reflection.
In this little volume, the author weaves a delightful tapestry using this result as his touchstone. He treats reflection principles, Plemelj formulas, analytic continuation, Dirichlet problems, Neumann problems, and Cauchy problems. He looks at both the one- and several-variable theories. He treats classical results on reproducing formulas in a new way.
In all, the reader is treated (in a scant 96 pages) to a panorama of important issues in modern analysis. Since reflection principles are currently enjoying a surge in popularity, the book should be of particular interest.

30E99 Miscellaneous topics of analysis in the complex plane
30C40 Kernel functions in one complex variable and applications
30E25 Boundary value problems in the complex plane
46E20 Hilbert spaces of continuous, differentiable or analytic functions