  We prove the universality of the generalized QDD_{N1,N2} (quadratic dynamical
decoupling) pulse sequence for near-optimal suppression of general single-qubit
decoherence. Earlier work showed numerically that this dynamical decoupling
sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence
using N1 and N2 pulses respectively, can eliminate decoherence to O(T^N) using
O(N^2) unequally spaced "ideal" (zero-width) pulses, where T is the total
evolution time and N=N1=N2. A proof of the universality of QDD has been given
for even N1. Here we give a general universality proof of QDD for arbitrary N1
and N2. As in earlier proofs, our result holds for arbitrary bounded
environments. Furthermore, we explore the single-axis (polarization) error
suppression abilities of the inner and outer UDD sequences. We analyze both the
single-axis QDD performance and how the overall performance of QDD depends on
the single-axis errors. We identify various performance effects related to the
parities and relative magnitudes of N1 and N2. We prove that using QDD_{N1,N2}
decoherence can always be eliminated to O(T^min[N1,N2]).
