  We prove norm estimates for the difference of resolvents of operators and
their discrete counterparts, embedded into the continuum using biorthogonal
Riesz sequences. The estimates are given in the operator norm for operators on
square integrable functions, and depend explicitly on the mesh size for the
discrete operators. The operators are a sum of a Fourier multiplier and a
multiplicative potential. The Fourier multipliers include the fractional
Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are
real, bounded, and H\"older continuous. As a side-product, the Hausdorff
distance between the spectra of the resolvents of the continuous and discrete
operators decays with the same rate in the mesh size as for the norm resolvent
estimates. The same result holds for the spectra of the original operators in a
local Hausdorff distance.
