The pseudoanalytic extension.

*(English)*Zbl 0795.30034The paper under review is a survey. The main technique is a description of smooth functions in terms of the so-called analytic continuation. The idea of pseudoanalytic continuation was developed by the author. Roughly speaking the approach of pseudoanalytic continuation describes classes of various smooth functions as functions which admit a continuation to a neighbourhood of their domain such that the \(\overline\partial\) derivative is small near the boundary. This approach is shown to be very powerful. The author describes many classes of functions in terms of pseudoanalytic continuation: Hölder classes, Sobolev classes, Besov classes, Carleman classes.

In the survey under review the author demonstrates the technique of pseudoanalytic continuation by solving several problems or giving new proofs: Denjoy-Carleman theorem, interpolation problems in spaces of smooth functions, multipliers properties of inner functions on Hölder classes, and others.

The paper is written very clearly.

In the survey under review the author demonstrates the technique of pseudoanalytic continuation by solving several problems or giving new proofs: Denjoy-Carleman theorem, interpolation problems in spaces of smooth functions, multipliers properties of inner functions on Hölder classes, and others.

The paper is written very clearly.

Reviewer: V.V.Peller

##### MSC:

30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |

26E10 | \(C^\infty\)-functions, quasi-analytic functions |

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