Approximation by polyanalytic functions in Hölder spaces

The problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces C m C^m , m ∈ ( 0 , 1 ) m\in (0,1) , is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not lead to the triviality of the approximation conditions. In the model case of polyanalytic functions of order 3, approximation conditions and a constructive approximation method generalizing the Vitushkin localization method are studied. Some unsolved problems are formulated.