Beurling-Pollard type theorems

We establish a version of the Beurling–Pollard theoremfor operator synthesis and apply it to derive some results onlinear operator equations and to prove a Beurling–Pollardtype theorem for Varopoulos tensor algebras. In addition, weestablish a Beurling–Pollard theorem for weighted Fourieralgebras and use it to obtain ascent estimates for operatorsthat are functions of generalized scalar operators.