Local support, affine cancellation, and normalization of multilevel D4 detail rows

Expanding the recursion shows that the detail row at stage $\mathrm{\ell }$, translated to its first input index, has taps at indices $t=2^{\mathrm{\ell }-1}r+{\sum }_{v=1}^{\mathrm{\ell }-1}{2}^{v-1}{s}_v$, where each of $r,s_1,\dots ,s_{\mathrm{\ell }-1}$ ranges over $\{0,1,2,3\}$; each tuple contributes $g_r{\prod }_{v=1}^{\mathrm{\ell }-1}{h}_{s_v}$ at that index, and the effective coefficient at $t$ is the sum of these products over all tuples with that index. This follows by substituting the low-pass sum at each earlier stage into the current high-pass sum. The largest such index is $3(2^{\mathrm{\ell }}-1)$, so the row is supported in at most $3(2^{\mathrm{\ell }}-1)+1$ consecutive locations. Because $\mathrm{\ell }\le J-2$, this number is $3\cdot 2^{\mathrm{\ell }}-2\le 3\cdot 2^{J-2}-2<N$, so the locations are distinct modulo $N$.

Regard the effective taps in step 1.1 before reducing their input positions modulo $N$. A low-pass stage maps an affine input sequence $p(n)=\alpha +\beta n$ to an affine sequence: ${\sum }_{s=0}^3{h}_{s}p(2k+s)=2\beta k\sum h_s+\alpha \sum h_s+\beta \sum s{h}_s$, which is affine in $k$. Repeating this argument keeps the coarse sequence affine. The next high-pass stage applied to an affine sequence $A+Bk$ gives $A\sum g_s+B(2k\sum g_s+\sum s{g}_s)=0$ by [F2]. Therefore the effective detail tap row has zero sum and zero first moment. After summing all tuple contributions at each lifted location, periodic reduction assigns the resulting effective coefficients to residues. Step 1.1 confines these locations to one interval of fewer than $N$ consecutive integers, so reduction modulo $N$ is injective on that interval and the row lifts back to the same interval.

By [F3], each row of the full multilevel analysis matrix has squared Euclidean norm $1$. If its support has $S$ locations with coefficients $z_1,\dots ,z_S$, then $({\sum }_{i=1}^S\mathrm{\mid }{z}_i\mathrm{\mid }{)}^2\le S{\sum }_{i=1}^S\mathrm{\mid }{z}_i{\mathrm{\mid }}^2=S$, since $S{\sum }_i\mathrm{\mid }{z}_i{\mathrm{\mid }}^2-({\sum }_i\mathrm{\mid }{z}_i\mathrm{\mid }{)}^2={\sum }_{i<j}(\mathrm{\mid }{z}_i\mathrm{\mid }-\mathrm{\mid }{z}_j\mathrm{\mid }{)}^2\ge 0$. Hence its ${\mathrm{\ell }}^1$ norm is at most $\sqrt{S}$. The detail block at stage $\mathrm{\ell }$ has the length of the current approximation after downsampling, namely $N\mathrm{/}{2}^{\mathrm{\ell }}=2^{J-\mathrm{\ell }}$ rows. This and the affine cancellation from step 2.1 establish every clause of the Statement.