Some perturbation results for quasi-bases and other sequences of vectors

We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space $\text{Hil}$ and producing new sequences which share, with the original ones, { reconstruction formulas on a dense subspace of $\text{Hil}$ or on the whole space}. We also propose some preliminary results on the same issue, but in a distributional settings.