Let $R$ be a commutative Noetherian ring, $\!M$ an $\!R$-module and $d$ a non-negative integer. Let $\Sigma$ denote the set of ideals $\frak{I}$ of $R$ such that $\mathrm{dim}(R/\frak{I})\!\leq\!d$. For an ideal $\frak{b}$ of $R$, we define the $(d,\i\frak{b})$-transform $D_{d,\i\i\frak{b}}(M)$ and study its properties. Then a criterion for $D_{d,\frak{b}}(R)\!=\!\bigcap_{\frak{p}\notin W(d,\frak{b})}R_{\frak{p}}$ will be given, where $W(d,\frak{b})$ contains all ideals $\frak{a}$ of $R$ such that $\frak{I}\subseteq \frak{a}+\frak{b}$ for some $\frak{I}\in \Sigma$. For each $i\geq 0$, let $D^i_{d,\frak{b}}(-)$ denote the $i$-th right derived functor of $D_{d,\frak{b}}(M)$. We study the localization of the module $D^i_{d,\frak{b}}(M)$ and prove that  $D^i_{d,\frak{b}}(M)_\frak{p}\cong D^i_{d-\textrm{dim}(R/(\frak{p}+\frak{b})),\frak{b}_\frak{p}}(M_\frak{p})$ for all $\frak{p}\in\mathrm{Spec}(R)$ and all $i\geq 0$. Finally, we establish vanishing theorems for $D^i_{d,\frak{b}}(M)$.