  The work here enables quasi-linear cost-scaling with $t$ while keeping ${\rm
polylog} (1/\epsilon)$ scaling and no extra block-encoding qubits, where
$\epsilon$ is the algorithmic error. This work opens up the possibility for
off-loading signal processing with the use of low-weight Fourier expansion
which avoids the Gibbs phenomenon and has low $1$-norm on the coefficients.
