Convolution operators on the disk

Abstract. In this work, we introduce hyperbolic convolution operators using a subgroup of the Blaschke group. The subgroup in question can be interpreted as the group of direction preserving translations on the complex unit disk. We introduce a group norm and the Haar integral which are invariant with respect to this hyperbolic translation. Using these tools, we propose the notions of hyperbolic convolution and approximate identity. Finally, we extend the investigated group of direction preserving hyperbolic translations with a dilation operation and investigate some properties of the resulting field.