  We establish a version of the Beurling-Pollard theorem for operator synthesis
and apply it to derive some results on linear operator equations and to prove a
Beurling-Pollard type theorem for Varopoulos tensor algebras. Additionally we
establish a Beurling-Pollard theorem for weighted Fourier algebras and use it
to obtain ascent estimates for operators that are functions of generalized
scalar operators.
