  We prove a uniform generalized gaussian bound for the powers of a discrete
convolution operator in one space dimension. Our bound is derived under the
assumption that the Fourier transform of the coefficients of the convolution
operator is a trigonometric rational function, which generalizes previous
results that were restricted to trigonometric polynomials. We also allow the
modulus of the Fourier transform to attain its maximum at finitely many points
over a period.
