Polyanalytic Besov spaces and approximation by dilatations

Using partial derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\text{ol}z}f$, we introduce Besov spaces of polyanalytic functions on the unit disk and on the upper half-plane. We then prove that the dilatations of each function in polyanalytic Besov spaces converge to the same function in norm. This opens the way for the norm approximation of functions in polyanalytic Besov spaces by polyanalytic polynomials.