  We study transformations of finite modules over Noetherian local rings that
attach to a module $M$ a graded module $H^{0}_{\mathfrak{m}}(
\mathrm{gr}_{I}(M))$ defined via partial systems of parameters of $M$. Despite
the generality of the process, which are called $\mathbf{j}$-transforms, in
numerous cases they have interesting cohomological properties. We focus on
deriving the Hilbert functions of $\mathbf{j}$-transforms and studying the
significance of the vanishing of some of its coefficients.
