its derivatives, and convolution products resulting from points inside of a cube. These estimates are used in Definition \[def:KrawczykApprox\] to construct an outer approximation to the Krawczyk operator. The reader is encouraged to skip the proof of Lemma \[prop:CentralLemma\] on a first reading, which is best summarized as bounding various infinite sums by various finite sums and the estimate in . These bounds are presented in Definition \[def:Gfunctions\], all of which are given as a finite number of operations, explicitly computable in terms of $C_0$ and the $\pi'_M$-projection of a given cube. In Lemma \[prop:DiscreteConv\] we define the constant $\gamma_M$ which is needed for the definition of .

\[prop:DiscreteConv\] Let $s \geq 2$ and let $s_*$ be the largest integer such that $ s_* \leq s$ and define: $$\gamma_k := 2 \left[ \frac{k}{k-1}\right]^s + \left[ \frac{4 \ln (k-2)}{k} + \frac{\pi^2 -6}{3} \right] \left[ \frac{2}{k} + \frac{1}{2} \right]^{s_*-2}.$$ For $k \geq 4$, we have that $  \sum_{k_1 =1}^{k-1} \frac{k^s}{k_1^{s} (k-k_1)^s} \leq  \gamma_k $. If $ 6 \leq M \leq k$, then $ \gamma_k \leq  \gamma_M $.

\[def:Gfunctions\] Fix a cube $ X  $ with $ s >2$, define $C_1 := \sup_{x\in X} \| \pi_c x \|_s$, and select a point $ 