Norm Approximation by Polynomials in Some Weighted Bergman Spaces

The polynomials are shown to be dense in weighted Bergman spaces in the unit disk whose weights are superbiharmonic and vanish in an average sense at the boundary. This leads to an alternative proof of the Aleman–Richter–Sundberg Beurling-type theorem for zero-based invariant subspaces in the classical Bergman space. Additional consequences are deduced.