Trofymenko O. Mean value theorems for polyanalytic functions and their generalizations

This thesis is devoted to the mean value theorems for solutions of homogeneous linear partial differential equations with constant coefficients in the complex plane whose left hand side is represented in the form of the product of some non-negative integer powers of the formal Cauchy derivatives. We consider systems of special type homogeneous convolution equations defined on smooth functions in a disk, which generalize the classical mean value property over disks for harmonic functions. A sharp version of the uniqueness theorem for functions satisfying such a system in the case of one equation has been established. We also investigate the case of two equations and prove a theorem consisting the classical Delsarte's two-radii theorem for dimension two as a special case.