Research proof index

Let $u_1\ne u_2$ be forest tops with $\mathcal I(u_1)\le\mathcal I(u_2)$. If a tile $p$ belongs to either tree and its spatial cube overlaps $\mathcal I(u_1)$, then the two top frequencies are separated at the scale of $p$ by

$2^{Zn/2}\le d_p(\mathcal Q(u_1),\mathcal Q(u_2)).$

Fix a cube $J$ in the forest’s remaining-cube family. Consider grid cubes $I$ at relative scale $s(J)-s'$, disjoint from the top cube $\mathcal I(u_1)$, whose enlarged balls meet $J$. The normalized average over $J$ of the square of the number of those enlarged balls containing each point is bounded by the scale-dependent constant $C(a,s')$.

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 $x'\in L$.

Product rule for the time derivative of a product of matrix-valued functions.

The time derivative commutes with transpose.

The matrix reps of φ and f φ agree.