Unit 3, continued: Projection increments and smooth mean-zero rows

This part studies Corollary 4.5 and Lemma 4.6 in D §§4.3–4.4, pp.23–26. The preceding part proved an abstract row estimate for varying functions. We now verify its assumptions for two specific classes of operators.

For projection increments, orthogonality directly proves the joint Bessel bound. For smooth rows, the original Bessel bound remains an explicit hypothesis; the argument shows how to apply it to functions at different depths. Both estimates enter the projection construction and delayed replacement in D §5, and ultimately the local absolute-sum estimate of Theorem 3.2.

1. The varying-function estimate used below

Let $R$ be a finite dyadic root interval of positive length, with spatial depth $s_R$. The space $L^2(R)$ uses ordinary Lebesgue measure. Complex Hilbert-space inner products are linear in their first variable.

The function-index set $\mathcal S$ is a finite interval of integers, partitioned into $W$ deterministic nonempty consecutive blocks. The strongly measurable functions $F_s:R\to\mathcal H$ satisfy on each block

D Lemma 4.4 applies to finitely many row operators $A_\nu$ arranged in rounds. Each row is supported on a whole interval of that round's support partition. Its adjoint is a constant-coefficient combination of fixed column functions on the observation partition. The observation partition refines the current support partition, and the next support partition refines the preceding observation partition. All rows satisfy a joint Bessel bound for arbitrary fixed functions. Every row in a round uses the same $F_{s_n}$, with $s_n$ nondecreasing. If the fixed columns satisfy the preceding private-coordinate conditions with constant $D\ge2$ and the Bessel constant is $B$, then

Each application below checks these hypotheses. An estimate holding only for the function used in the construction is not treated as a joint operator bound.

2. Inherited projections and nesting

Source: D Corollary 4.5, p.23.

Retain the fixed index set and columns $\Phi_v$. Each index has a birth interval; restriction to descendants changes neither its function nor its private direction. Along each spatial path, at most $D$ indices are available, column norms are at most $D$, and private coefficients have absolute value at least $D^{-1}$.

Fix a finite consecutive range of integer depths satisfying $h\ge s_R$ and define all projections in that range. Let $\mathcal P_h$ be the full partition of $R$ at depth $h$. For each $Q\in\mathcal P_h$, choose a subset $M_h(Q)$ of available indices. Define $P_{h,Q}$ as the orthogonal projection from $L^2(Q;\mathcal H)$ onto

This space is finite-dimensional and hence closed. Let $P_h$ be the direct sum of these local projections on $L^2(R;\mathcal H)$.

Assume inheritance: for every child interval $Q'\in\mathcal P_{h+1}$ with $Q'\subseteq Q$,

If $z$ belongs to the range of $P_h$, it is such a constant-coefficient combination on every $Q$. Restrict the same coefficients to each child. By (Inheritance), the result belongs to the next local range. Thus

For $r\le t$, the two orthogonal projections satisfy

The difference $P_t-P_r$ is self-adjoint and idempotent, hence is itself an orthogonal projection.

3. Ordered projection increments satisfy a joint Bessel bound

Take finitely many depth intervals and function indices satisfying

The endpoints lie in the range where the projections are defined. Function indices need not equal projection depths.

Write $\Delta_j=P_{t_j}-P_{r_j}$. If $j<l$, then $t_j\le r_l$, and the nested-projection identities give directly

The increment ranges are therefore pairwise orthogonal. Their sum $\sum_j\Delta_j$ is an orthogonal projection, so

3.1 Rows on whole support intervals

For every $Q\in\mathcal P_{r_j}$, define

The local partitions for $P_{r_j}$ and $P_{t_j}$ both refine $\mathcal P_{r_j}$, so both projections commute with multiplication by $\mathbf1_Q$. The row input and output are therefore supported on $Q$, and

Sum over $j$. Equation (Bessel-increments) says exactly that all these rows retain Bessel constant $B=1$. This is a joint operator bound for every $z$.

3.2 The observation partition for the adjoint

We have $A_{j,Q}^*=\mathbf1_Q\Delta_j$. On an interval of depth $t_j$, the output of $P_{t_j}$ is a constant-coefficient combination of available columns. The output of $P_{r_j}$ is such a combination on its ancestor interval. After restriction, inheritance makes its columns available on the descendant as well.

In round $j$, take support partition $\mathcal P_{r_j}$ and observation partition $\mathcal P_{t_j}$. The relation $t_j\ge r_j$ gives refinement within the round, and $r_{j+1}\ge t_j$ gives refinement between rounds. All rows in the round use $F_{s_j}$, with nondecreasing indices. Apply (Row-bound) with $B=1$ to obtain the paper's (4.11):

3.3 Depth intervals with bounded overlap

If the intervals split into $m$ subfamilies ordered as above, each with nondecreasing function indices, apply (Projection-sum) separately and add. The right side becomes $mC_D^2|R|$.

A sufficient condition is that, after zero increments $r=t$ are removed, every integer edge is covered by at most $m$ intervals and the function indices are nondecreasing in left-endpoint order. Color greedily in that order, assigning an interval a color whose preceding endpoint is no greater than its start. If all $m$ colors were occupied by unfinished intervals, those intervals and the new one would cover the first edge after the current start, contradicting overlap at most $m$. Thus $m$ colors suffice, each with the required ordered intervals and indices.

4. Single-step and delayed increments

Source: D (4.12)–(4.13), pp.23–24.

For every $s\in\mathcal S$ and $s\ge s_R$, let $\mathcal U_s$ be any subset of the depth-$s$ intervals. Fix an integer $\ell$ and impose the condition that both $s+\ell,s+\ell+1$ lie in the projection range for every retained $s$. The intervals

are ordered as $s$ increases, and the function index is also $s$. By (Projection-sum),

For fixed $s$, the intervals $I\in\mathcal U_s$ are pairwise disjoint, and the normalized norms satisfy $|I|\|z\|_{2,I}^2=\|\mathbf1_Iz\|_2^2$. Restricting the already estimated outputs decreases the square sum. Hence

The estimate is applied first to whole rows, then the outputs are restricted. We have not arbitrarily split a supported row and asserted that the pieces retain the original structural hypotheses.

Fix an integer $L\ge0$ and keep only indices for which $s,s+L$ all lie in range. For $L>0$ there is the exact telescoping identity

Use Minkowski's inequality in the Hilbert direct sum indexed by $(s,I)$, and apply (Unit-step) for each $\ell$. This yields

For $L=0$ both sides vanish. As the projection scale changes, every term still uses the same $F_s$; it is not replaced by $F_{s+L}$.

5. The full statement for smooth rows with fixed multipliers

Source: D Lemma 4.6, pp.24–26.

Take a finite index set $\mathcal N$. For every $\nu\in\mathcal N$, fix a measurable function $m_\nu:R\to\mathbb C$ satisfying $\|m_\nu\|_\infty\le1$. All intervals bearing the same $\nu$ must use the same $m_\nu$. It may oscillate arbitrarily; neither continuity nor frequency separation is assumed.

Choose a finite set $\mathcal E$ of pairs $(I,\nu)$, where $I\subseteq R$ is dyadic of depth $s(I)$. Each pair occurs at most once. Let $\omega_{I,\nu}\in C^1(I)$ satisfy

where $A\ge0$. Define the physical scalar row

Assume there is $B_H\ge0$ such that these selected rows satisfy, for every $f\in L^2(R)$,

This assumption concerns only $\mathcal E$. It need not remain true for a larger family formed from every possible interval–multiplier pair.

Let the scalar functions $f_s$ satisfy (Variation), with every $s(I)\in\mathcal S$. Set

The paper's equation (4.14) then states

The proof preserves the fixed-column form of the adjoints in every round and bounds the Bessel constant contributed by the added coordinates.

6. Martingale differences and complex conjugation

If $\mathcal E$ is empty, the conclusion is zero. Otherwise extend every $\omega_{I,\nu}$ by zero outside $I$. Let $\mathbb E_h$ be conditional expectation onto the full depth-$h$ partition, taking the mean on each small interval.

Fix a row and write $s=s(I)$. Mean zero gives $\mathbb E_s\overline\omega_{I,\nu}=0$. Since the inner product is linear in its first variable, define

and

The conjugate of $\omega$ is essential. The integral in (Smooth-row) contains no conjugation, so its Hilbert-space representation is

The conjugates can be checked directly on p.25 of the original PDF; text extraction may omit their bars.

Each $q_{I,\nu,h}$ is supported in $I$ and constant on intervals of depth $s+h$. Zero extension creates no boundary issue because each small interval lies inside $I$ or is disjoint from it.

6.1 Norm decay from the derivative bound

A parent interval at depth $s+h-1$ has length $|I|2^{-(h-1)}$. The difference between a child's mean and its parent's mean is bounded by the function's oscillation on that parent. Thus

Hence

For each fixed $\nu,h$, these functions are pairwise orthogonal as $I$ varies. Different support depths belong to orthogonal martingale-difference spaces. At equal depth, distinct dyadic intervals are disjoint, and each pair $(I,\nu)$ occurs at most once. By Bessel, the analysis operator for fixed $\nu,h$ has norm at most $2A2^{-h}$.

6.2 Where the fixed multiplier is used

The physical row at level $h$ is $f\mapsto\langle m_\nu f,q_{I,\nu,h}\rangle$. For each fixed $\nu$, all rows act first on the same function $m_\nu f$, and multiplication has operator norm at most one. Therefore

Sum over $\nu$ to bound the whole level's physical row-operator norm by

If one index used different multipliers on different intervals, these orthogonal tests would no longer act on a common $m_\nu f$. This part of the proof would fail.

7. The initial Bessel bound is inherited from the selected row family

The derivative bound makes $\mathbb E_{s+H}\overline\omega$ converge uniformly on $I$ to $\overline\omega$. Thus

Summability of (Physical-tail) implies convergence of the physical row-operator series in operator norm. Subtracting the tail from the original row operator gives

The last step uses $2^{-h_0}\le(2d)^{-10}$ and $d\ge2$. The original assumption (Physical-Bessel) is genuinely needed; smoothness alone has not been used to infer a joint Bessel bound for the original family.

8. One private direction for each fixed multiplier

Set

The columns are fixed on the root $R$ from the start and only restricted thereafter. Their norms are at most $\sqrt2$, their private coefficients equal one, and each path carries $\#\mathcal N\le d$ indices. The preceding column hypotheses hold with constant $D=d$.

For $h=0$ or $h>h_0$, define the augmented scalar row

Its adjoint maps the scalar $c$ to $c\,q_{I,\nu,h}\Phi_\nu$. For the physical function $Z=(f,0)$,

which agrees with part $h$ of the original row. The first column component must be $\overline m_\nu$ to match the inner-product convention.

8.1 The augmented tail-operator bound

For a private function $z=(z_\nu)$, level $h$ has output $\langle z_\nu,q_{I,\nu,h}\rangle$. Orthogonality for fixed $\nu$ and the orthogonal direct sum across private coordinates bound its operator norm by $2A2^{-h}$.

The physical and private functions occupy orthogonal domain components. For any $(f,z)$, apply the two operator bounds followed by two-dimensional Cauchy–Schwarz:

Consequently,

8.2 The augmented initial-operator bound

The initial private part sums the first $h_0$ martingale-difference analysis operators and has norm at most

The physical part is bounded by $B_H+A$ in (Physical-initial). The operator triangle inequality gives

Every operator here uses the same originally selected row-index set $\mathcal E$. No extra Bessel hypothesis on a larger family is invoked.

9. Support and observation partitions arranged by depth residue classes

Source: D p.26.

The initial observation depth is $s+h_0$; tail part $h$ has observation depth $s+h$. Write both as $s+\ell$. Each row is supported on a whole $I$, and its adjoint coefficient $q$ is constant on intervals of depth $s+\ell$.

Group the selected support depths by residue modulo $\ell+1$. Within one residue class, put every row of the same depth $s$ into one round. Use the full depth-$s$ partition for supports and the full depth-$s+\ell$ partition for observations. Unselected intervals remain in the partitions, with no rows placed on them.

Successive support depths differ by at least $\ell+1$, so the next support partition refines the preceding observation partition. All rows in a round use $F_s=(f_s,0)$, with indices increasing across rounds. Restriction to one residue class only removes outputs, retaining the joint bound (Augmented-tail) or (Augmented-initial). Every hypothesis of (Row-bound) is therefore satisfied.

Let $V_0\in\ell^2(\mathcal E)$ be the initial outputs on their respective $f_{s(I)}$, and let $V_h$ be the outputs of tail part $h$. Extend each residue-class vector by zero into the common $\ell^2(\mathcal E)$ and use the triangle inequality to obtain

and

At most $\ell+1$ residue classes are used. The paper's constants do not require an improvement of this factor.

10. Summing the tail and checking the constants

The rowwise uniform expansion identifies the original output vector with

The norm bounds on the right are summable, so the series converges in $\ell^2(\mathcal E)$. Differentiating a geometric series, or calculating directly, gives

Minkowski's inequality proves the first line of (Smooth-estimate).

Set $l=\log_2(2d)\ge2$. Since $h_0=\lceil10l\rceil$,

The quantity in parentheses is therefore at most

Multiplication by $C_d\le(40/3)l$ gives $57\cdot40/3=760$, the second line of (Smooth-estimate). The constant is independent of the number of selected intervals, maximal depth, and function-block count $W$.

Scope and subsequent use

• D pp.23–24, Corollary 4.5, equations (4.11)–(4.13): inherited projection increments, whole-support rows, bounded overlap, and delayed projections.
• D pp.24–26, Lemma 4.6, equations (4.14)–(4.18): smooth mean-zero rows with fixed multipliers, martingale differences, added coordinates, residue classes, and tail summation.

The next step, D §§5.1–5.2, checks how projections from local detection satisfy these hypotheses and why cutting original intervals when the count limit is exceeded loses at most one quarter of the root length. The two estimates here do not yet prove D Theorem 3.2. That theorem also depends on the constructions, frequency counts, and common parameter choices of §§5–7, and the individually specified analytic inputs from H.