  We study the problem of maintaining a differentially private decaying sum
under continual observation. We give a unifying framework and an efficient
algorithm for this problem for \emph{any sufficiently smooth} function. Our
algorithm is the first differentially private algorithm that does not have a
multiplicative error for polynomially-decaying weights. Our algorithm improves
on all prior works on differentially private decaying sums under continual
observation and recovers exactly the additive error for the special case of
continual counting from Henzinger et al. (SODA 2023) as a corollary.
  Our algorithm is a variant of the factorization mechanism whose error depends
on the $\gamma_2$ and $\gamma_F$ norm of the underlying matrix. We give a
constructive proof for an almost exact upper bound on the $\gamma_2$ and
$\gamma_F$ norm and an almost tight lower bound on the $\gamma_2$ norm for a
large class of lower-triangular matrices. This is the first non-trivial lower
bound for lower-triangular matrices whose non-zero entries are not all the
same. It includes matrices for all continual decaying sums problems, resulting
in an upper bound on the additive error of any differentially private decaying
sums algorithm under continual observation.
  We also explore some implications of our result in discrepancy theory and
operator algebra. Given the importance of the $\gamma_2$ norm in computer
science and the extensive work in mathematics, we believe our result will have
further applications.
