Unit 3, continued: delaying the projection at a fixed depth of use

This part develops D Lemma 5.5, pp.35–37, equations (5.18)–(5.22). Fix one triple from the predictor family in the preceding part. The aim is to replace the actual function $F_{a,s}$ by the finer projection $P_{a,s+K_0}F_{a,s}$ . Every refinement acts on the same $F_{a,s}$ .

The delay error has two sources. Variation of the smooth test function on refined intervals contributes $2^{-K_0}$ . The predictor's private coordinate contributes $\sqrt\epsilon$ , and crossing $K_0$ projection levels gives $K_0\sqrt\epsilon$ . We derive these quantities from orthogonality and a finite telescoping identity.

1. The full assumptions and conclusion

Fix the two complementary positions $a,b$ and an index $\alpha$ from Corollary 5.4 in the preceding part. Abbreviate

\psi_a=\psi_{a,\alpha},\quad \psi_b=\psi_{b,\alpha},\quad \lambda=\lambda_\alpha.

The two functions are fixed and satisfy $\|\psi_a\|_\infty,\|\psi_b\|_\infty\le1$ . Inside a dyadic subroot $K\subseteq J'$ , choose a finite family $\mathcal U$ of intervals of use, retaining only those with

|\lambda||I|^2\le D_{\mathrm{pred}}.

Position $a$ has inherited orthogonal projections $P_{a,h}$ on complete partitions, with a fixed column-function constant $D_h$ . Here $h$ is the projection depth. The symbol $D_h$ denotes one numerical bound for the column family in the complementary position, rather than a depth-indexed sequence of bounds.

Fix an integer $K_0\ge0$ and a number $0<\epsilon\le1$ . Suppose that on every interval of use and all its further descendants, the projection family contains the same predictor column

\Phi_v=(\overline{\psi_a},\sqrt\epsilon\,e_v). \tag{Predictor-column}

The predictor column is included in the column-function bound, so $\sqrt\epsilon\ge D_h^{-1}$ . Further columns may be added, subject to the same prescribed bound $D_h$ and with distinct private directions. The predictor column $v$ retains its own direction $e_v$ .

The original root is $J$ , and every depth $s(I)=\log_2(|J|/|I|)$ keeps its original index. The sequence $F_{a,s}=(f_{a,s},0)$ is the same complete normalized sequence restricted to $K$ . In position $b$ , the function $z_{b,I}$ is a sum of components from the three permitted opposing classes in the preceding part, with parameters $D_h,L$ .

The conclusion is

\begin{aligned} \sum_{I\in\mathcal U}|I|\, \bigl|\mathcal T_I^\alpha(F_{a,s(I)},z_{b,I}) -\mathcal T_I^\alpha(P_{a,s(I)+K_0}F_{a,s(I)},z_{b,I})\bigr| \\ \le C\bigl(2^{-K_0}+(1+K_0)\sqrt\epsilon\bigr) \operatorname{poly}\bigl(D_{\mathrm{pred}},\log(2D_h),1+L\bigr)|K|. \end{aligned} \tag{Delayed-error}

where the forms use only the physical scalar components of the functions.

2. The factor $\epsilon$ in residual means and $\sqrt\epsilon$ in private coordinates

Fix a depth of use $s$ . As the projection depth $h\ge s$ varies, keep $F_{a,s}$ fixed. Write

r_h^{[s]} =\bigl(f_{a,s}-(P_{a,h}F_{a,s})_{\mathrm{phys}}\bigr)\psi_a, \qquad \mu_h^{[s]}=\mathbb E_h r_h^{[s]},

Here $\mathbb E_h$ is conditional expectation on the complete partition of $K$ at depth $h$ .

Take a small interval at depth $h$ , denoted by $Q$ , inside an interval of use. Let the coefficient of the predictor column in $P_{a,h}F_{a,s}$ be $b_{h,v}^{[s]}$ . On $Q$ , the residual is orthogonal to (Predictor-column). Since the inner product is linear in the first variable,

0=\frac1{|Q|}\langle F_{a,s}-P_{a,h}F_{a,s},\Phi_v\rangle_{L^2(Q)} =\frac1{|Q|}\int_Qr_h^{[s]}-\epsilon b_{h,v}^{[s]}.

Hence

\mu_h^{[s]}=\epsilon b_{h,v}^{[s]}. \tag{Mean-coefficient}

Set $\Delta_h=P_{a,h+1}-P_{a,h}$ . The projection's private component in direction $v$ is $\sqrt\epsilon\,b_{h,v}^{[s]}$ , so on the finer intervals

\mu_{h+1}^{[s]}-\mu_h^{[s]} =\sqrt\epsilon\,(\Delta_hF_{a,s})_v, \tag{Mean-increment}

and

r_h^{[s]}-r_{h+1}^{[s]} =\psi_a(\Delta_hF_{a,s})_{\mathrm{phys}}. \tag{Residual-increment}

These three identities remain valid after further columns are added, since the same predictor column and its private direction are retained. The physical component of the column must be $\overline{\psi_a}$ , while the residual is multiplied by $\psi_a$ . This conjugation can be checked directly on p.35 of the source PDF.

3. A finite identity for residual moments

Take an interval of use $I$ at depth $s$ , and write $h_*=s+K_0$ . Let $\omega_I$ be a smooth function satisfying

\|\omega_I\|_\infty+|I|\|\omega_I'\|_\infty\le A_\omega.

For now, its integral need not be zero. For every finite $n>h_*$ , the source equation (5.20) reads

\begin{aligned} \int_Ir_{h_*}^{[s]}\omega_I ={}&\int_I\mu_{h_*}^{[s]}\omega_I\\ &+\sum_{h=h_*}^{n-1}\int_I (r_h^{[s]}-r_{h+1}^{[s]})(\omega_I-\mathbb E_h\omega_I)\\ &+\sum_{h=h_*}^{n-1}\int_I (\mu_{h+1}^{[s]}-\mu_h^{[s]}) (\mathbb E_{h+1}\omega_I-\mathbb E_h\omega_I)\\ &+\int_Ir_n^{[s]}(\omega_I-\mathbb E_n\omega_I). \end{aligned} \tag{Telescope}

To verify it, define

T_h=\int_Ir_h^{[s]}(\omega_I-\mathbb E_h\omega_I).

Direct subtraction gives

\begin{aligned} T_h-T_{h+1} ={}&\int_I(r_h^{[s]}-r_{h+1}^{[s]}) (\omega_I-\mathbb E_h\omega_I)\\ &+\int_Ir_{h+1}^{[s]} (\mathbb E_{h+1}\omega_I-\mathbb E_h\omega_I). \end{aligned}

The test function in the second term is measurable for the depth- $h+1$ partition, so $r_{h+1}^{[s]}$ may be replaced by $\mu_{h+1}^{[s]}$ . Moreover, $\mu_h^{[s]}$ is measurable for the depth- $h$ partition, so

\int_I\mu_h^{[s]} (\mathbb E_{h+1}\omega_I-\mathbb E_h\omega_I)=0.

These are the two terms in the telescoping sum. Finally,

\int_Ir_{h_*}^{[s]}\omega_I =\int_I\mu_{h_*}^{[s]}\omega_I+T_{h_*},

and summing $T_h-T_{h+1}$ over finitely many levels proves (Telescope). All integrals are complex bilinear pairings; these conditional-expectation identities require no additional conjugation.

4. The terminal term vanishes without convergence of projections to the original function

For $h\ge s$ , the complete projection partition refines the interval of use $I$ . Thus $P_{a,h}$ commutes with $\mathbf1_I$ . The contraction property of projection and the bound $|\psi_a|\le1$ give

\|r_h^{[s]}\|_{L^2(I)} \le2\|F_{a,s}\|_{L^2(I)}.

Smoothness gives

\|\omega_I-\mathbb E_n\omega_I\|_{L^2(I)} \le C A_\omega |I|^{1/2}2^{s-n}.

Cauchy–Schwarz therefore makes the terminal term in (Telescope) tend to zero.

If the original projections are defined only up to a finite depth, retain all existing columns and locally reproject on the finer intervals required. The column-function constant is unchanged. Apply the identity for each finite $n$ , then let $n\to\infty$ . The estimate does not assume $P_{a,n}F_{a,s}\to F_{a,s}$ .

5. Two projection-energy estimates across depths of use

These are D equation (5.21), p.36, following from the previously developed Corollary 4.5.

For each fixed integer offset $\ell$ , the intervals at depth of use $s$ are disjoint, so

\sum_s\sum_{\substack{I\in\mathcal U\\s(I)=s}} \|\mathbf1_I\Delta_{s+\ell}F_{a,s}\|_2^2 \le C\bigl(\log(2D_h)\bigr)^2|K|. \tag{Offset-energy}

Only valid projection depths are included. Similarly, for an integer $L_{\mathrm{proj}}\ge0$ ,

\sum_s\sum_{\substack{I\in\mathcal U\\s(I)=s}} \|\mathbf1_I(P_{a,s+L_{\mathrm{proj}}}-P_{a,s})F_{a,s}\|_2^2 \le C(1+L_{\mathrm{proj}})^2 \bigl(\log(2D_h)\bigr)^2|K|. \tag{Rolling-energy}

Both estimates follow by reapplying the full row estimate on the subroot $K$ . However fine the projection partition becomes, its operand remains $F_{a,s}$ from the depth of use.

6. Error rows for the smooth mean-zero part

First replace $\mathrm e(\lambda t^2)$ by one in the form. As in the preceding part, the resulting kernel has a tensor expansion in smooth, interior-supported functions with exactly zero mean in each variable. Fix a tensor index and apply (Telescope) to the test function in position $a$ , denoted by $\omega_I$ .

6.1 The two remainder sums contribute $2^{-K_0}$

For $h=s+\ell$ , smoothness gives

\|\omega_I-\mathbb E_h\omega_I\|_\infty +\|\mathbb E_{h+1}\omega_I-\mathbb E_h\omega_I\|_\infty \le C A_\omega2^{-\ell}.

The normalized test in the first remainder sum satisfies

|I|^{-1/2} \left|\int_I\psi_a(\Delta_{s+\ell}F_{a,s})_{\mathrm{phys}} (\omega_I-\mathbb E_{s+\ell}\omega_I)\right| \le C A_\omega2^{-\ell} \|\mathbf1_I\Delta_{s+\ell}F_{a,s}\|_2.

By (Mean-increment), the second remainder sum has an additional factor $\sqrt\epsilon\le1$ and obeys the same bound.

For fixed $\ell$ , apply (Offset-energy), then use Minkowski's inequality in the interval-indexed Hilbert direct sum to sum over $\ell\ge K_0$ . Since

\sum_{\ell\ge K_0}2^{-\ell}=2^{1-K_0},

the combined output norm of the two remainder sums is at most

C A_\omega2^{-K_0}\log(2D_h)|K|^{1/2}. \tag{Smooth-remainders}

6.2 The initial mean contributes $K_0\sqrt\epsilon$

Here $\omega_I$ has integral zero, while $\mu_s^{[s]}$ is constant on $I$ . Hence

\int_I\mu_{h_*}^{[s]}\omega_I =\int_I(\mu_{h_*}^{[s]}-\mu_s^{[s]})\omega_I.

Telescoping (Mean-increment) gives

\mu_{h_*}^{[s]}-\mu_s^{[s]} =\sqrt\epsilon\,((P_{a,s+K_0}-P_{a,s})F_{a,s})_v.

Using the exact $K_0$ form of Corollary 4.5 and the same normalized test, the initial term has Hilbert direct-sum norm at most

C A_\omega K_0\sqrt\epsilon\log(2D_h)|K|^{1/2}. \tag{Zero-mean-initial}

For $K_0=0$ , this term is exactly zero.

7. Pairing with the row in the other position

At a fixed tensor index, $|I|$ times the form equals the normalized residual moment in position $a$ times the normalized row value in position $b$ , multiplied by the fixed coordinate Jacobian factor and the tensor coefficient. Apply Cauchy–Schwarz over all intervals of use. Position $a$ is controlled by (Smooth-remainders) and (Zero-mean-initial). Position $b$ has three cases.

For an original function sequence, the full multiplier is the fixed function $\psi_b$ , with one smooth mean-zero row per interval of use. Its ordinary Bessel bound follows directly from the smooth-row proof in Unit 3: decompose the test function into martingale differences. The analysis operator for the $h$ -th part has norm at most $2A_\omega2^{-h}$ ; summing over $h\ge1$ gives at most $2A_\omega$ . Thus the single-multiplier case of D Lemma 4.6 applies and gives the square-sum row estimate for varying functions.

For a current projected function, first apply the H Compression operator statement recorded in the preceding part to this fixed single-multiplier row family, then apply D Lemma 4.4. Both the support and observation partitions are the complete partition at the current depth of use, with the same index $s$ . The cost is a fixed polynomial in $\log(2D_h)$ .

For an energy-type function $Z_{b,I}$ , Cauchy–Schwarz for one row gives

|I|^{-1/2}\left|\int_I\psi_b\omega_I Z_{b,I}\right| \le A_\omega\|Z_{b,I}\|_{L^2(I)}.

The given total square-energy bound then bounds the output norm by $A_\omega L|K|^{1/2}$ .

The derivative bounds grow only polynomially in the tensor index, so the rapidly decaying tensor coefficients allow absolute summation of these estimates. The mean-zero part is therefore bounded by

C\bigl(2^{-K_0}+K_0\sqrt\epsilon\bigr) \operatorname{poly}\bigl(\log(2D_h),1+L\bigr)|K|

The H Compression result used here remains an explicit external assumption.

8. The quadratic-phase difference need not have zero mean

It remains to treat the kernel

w_I(x,t)\bigl(\mathrm e(\lambda t^2)-1\bigr).

Fix a depth of use $s$ and set

x_s=|\lambda||I|^2,

This depends only on the depth, not on the interval of use at that depth. Only depths with $x_s\le D_{\mathrm{pred}}$ are retained.

After rescaling to a unit interval, every required fixed-order derivative bound for this difference kernel contains the factor

x_s\operatorname{poly}(D_{\mathrm{pred}}).

The one-variable functions in its ordinary smooth tensor expansion may now have nonzero mean. They still have polynomial derivative costs and rapidly decaying coefficients.

8.1 Ordinary projection orthogonality at fixed $s$

The operand stays fixed at $F_{a,s}$ throughout refinement. Ordinary orthogonality of the nested projections therefore gives

\sum_{h\ge s+K_0}\|\Delta_hF_{a,s}\|_2^2 \le\|\mathbf1_KF_{a,s}\|_2^2.

This is a square sum with $s$ fixed and $h$ varying. It differs from the varying-function estimates across depths of use in Section 5.

The intervals of use at one depth are disjoint. Apply the smooth decay $2^{s-h}$ to the two remainder sums in (Telescope), then Cauchy–Schwarz in $h$ . The output norm is at most

C A_\omega2^{-K_0}\|\mathbf1_KF_{a,s}\|_2.

8.2 The initial term with nonzero mean

We can no longer subtract $\mu_s^{[s]}$ . Instead, (Mean-coefficient) gives directly

\mu_{h_*}^{[s]} =\sqrt\epsilon\,(P_{a,h_*}F_{a,s})_v.

Disjointness at one depth and the contraction property of projection bound the initial test's output norm by

C A_\omega\sqrt\epsilon\,\|\mathbf1_KF_{a,s}\|_2.

The output norm of the test in the other position at this fixed depth is at most $C A_\omega(1+L)|K|^{1/2}$ , by the original function bound, local projection contraction, or the given energy bound. Normalization of the original function gives $\|\mathbf1_KF_{a,s}\|_2\le|K|^{1/2}$ . The contribution at this depth is therefore at most

x_s\operatorname{poly}(D_{\mathrm{pred}}) \bigl(2^{-K_0}+\sqrt\epsilon\bigr)(1+L)|K|.

8.3 A fixed unscaled $\lambda$ makes the scale sum finite

For $\lambda=0$ , the difference kernel vanishes. Otherwise, $x_s$ decreases by the factor $1/4$ as depth increases, so

\sum_{s:x_s\le D_{\mathrm{pred}}}x_s\le C D_{\mathrm{pred}}.

The same bound holds for every selected finite subset of depths. Summing the estimate from Section 8.2 controls the entire quadratic-phase difference. Adding the bound from Section 7 proves (Delayed-error).

This requires one fixed unscaled phase parameter $\lambda$ . Choosing unrelated values $\lambda_s$ at different depths would invalidate the geometric-series argument.

9. Successive replacements stay within the permitted classes

This is the final paragraph of D Lemma 5.5, p.37.

If position $b$ has already undergone delayed projection, then on an interval of use $I$ with $s=s(I)$ we have the exact identity

P_{b,s+K_0}F_{b,s} =P_{b,s}F_{b,s} +(P_{b,s+K_0}-P_{b,s})F_{b,s}.

The first term is a current projected component. Restrict the entire second term to $I$ . Equation (Rolling-energy) gives

\sum_{I\in\mathcal U} \left\|\mathbf1_I(P_{b,s(I)+K_0}-P_{b,s(I)})F_{b,s(I)}\right\|_2^2 \le C(1+K_0)^2\bigl(\log(2D_h)\bigr)^2|K|.

It is therefore a permitted energy-type component with parameter

L'=C(1+K_0)\log(2D_h)

Subtracting an old current projection leaves exactly three components with coefficients $1,1,-1$ , as allowed by the preceding definition.

The entire delayed difference must be treated as one energy component. Splitting it into $K_0$ opposing components would no longer justify a bound of three on the number of components. The exact decomposition above allows the same error estimate to be used for the two successive replacements.

Source coverage and subsequent use

D p.35, equations (5.18)–(5.20): the full error statement, the conjugated predictor column, residual moments, and the finite telescoping identity.
D p.36, equations (5.21)–(5.22): varying-function energy at a fixed offset, the mean-zero initial term, and the quadratic difference-kernel estimate at a fixed depth of use.
D p.37: geometric summation with a fixed phase and the three-component decomposition for successive replacements.

The next step, D §5.6, must choose $B,K_0,\epsilon$ , sum over all pairs of functions in the fixed-atom positions, and construct additional high-precision projections whose final column-family logarithmic size is compatible with these errors. The present estimate says that once the parameters and an actual inherited projection family satisfy the stated assumptions, delayed replacement has the uniform error displayed above.