Research proof index

For a tree in the forest, a point $x$ in a boundary cube $L$, and a bounded compactly supported function $f$, the oscillatory sum of the error $f-\operatorname{approxOnCube}(f)$ over the relevant scales is bounded pointwise by a constant times the tree’s boundary operator applied to the cube-wise approximation of $|f|$, evaluated at any other point $x'\in L$.