  In \cite{gmw2022}, Guan, Murugan and Wei established the equivalence of the
classical Helmholtz equation with a ``fractional Helmholtz" equation in which
the Laplacian operator is replaced by the nonlocal fractional Laplacian
operator. More general equivalence results are obtained for symbols which are
complete Bernstein and satisfy additional regularity conditions. In this work
we introduce a novel and general set-up for this Helmholtz equivalence problem.
We show that under very mild and easy-to-check conditions on the Fourier
multiplier, the general Helmholtz equation can be effectively reduced to a
localization statement on the support of the symbol.
