This paper addresses the formal specification and verification of fast Fourier transform (FFT) algorithms at different abstraction levels based on the HOL theorem prover. We make use of existing theories in HOL on real and complex numbers, IEEE standard floating-point, and fixed-point arithmetics to model the FFT algorithms. Then, we derive, by proving theorems in HOL, expressions for the accumulation of roundoff error in floating- and fixed-point FFT designs with respect to the corresponding ideal real and complex numbers specification. The HOL formalization and proofs are found to be in good agreement with the theoretical paper-and-pencil counterparts. Finally, we use a classical hierarchical proof approach in HOL to prove that the FFT implementations at the register transfer level (RTL) implies the corresponding high level fixed-point algorithmic specification.