Reliable implementation of digital filters in finite-precision is based on accurate error analysis. However, a small error in the time domain does not guarantee that the implemented filter verifies the initial band specifications in the frequency domain. We propose a novel certified algorithm for the verification of a filter's transfer function, or of an existing finite-precision implementation. We show that this problem boils down to the verification of bounds on a rational function, and further to the positivity of a polynomial. Our algorithm has reasonable runtime efficiency to be used as a criterion in large design space explorations. We ensure that there are no false positives but false negative answers may occur. For negative we give a tight bound on the margin of acceptable specifications. Numerical results illustrate the application of our approach to comparison of transfer function design tools and of the various finite-precision implementations.