Unit 3, continued: summing scales after separation and fixing the predictor family

This part treats D §§5.3–5.4, pp.30–34, Lemma 5.3 and Corollary 5.4. The preceding quadrilinear expansion leaves two projection differences and two residual positions. We first fix two scalar structured functions from the projection differences and construct a family of functions for subsequently approximating the residuals. This family must be chosen before the functions in the other two positions and their projection constructions.

The argument uses scalar separation, uniform truncation, frequency counting, and fixed-function Bessel estimates from H. We identify these statements individually and explain how D applies them to depth-varying functions. The scalar results from H remain explicit assumptions here.

1. The approximation problem after fixing two positions

Among the four positions $\{0,1,2,3\}$ , choose distinct positions $i,j$ , and let $a,b$ be the other two. Write $y_e=x+et$ . Fix two bounded quadratic structured functions $\phi_i,\phi_j$ on a common initial root $J'$ . If they come from two existing columns, their common root is the deeper of the two initial intervals; disjoint initial intervals have no common interval of use.

Define the bilinear form

\mathcal L_I^{\phi_i,\phi_j}(z_a,z_b) =|I|^{-2}\int_{\mathbb R^2}w_I(x,t) \phi_i(y_i)\phi_j(y_j)z_a(y_a)z_b(y_b) \,\mathrm{d}x\,\mathrm{d}t.

For now, keep the scalar coefficients in the projection expansion outside this form, separate from the fixed structured functions.

The progression separation in H Lemma 4.5 (p.20) expresses the product of the two structured functions in the fixed-atom positions (the paper's low positions) as an absolutely controlled sum of separated terms. After its scalar coefficient is removed, each term has the form

\psi_a(y_a)\psi_b(y_b)Q(t), \quad \|\psi_a\|_\infty,\|\psi_b\|_\infty,\|Q\|_\infty\le1.

These three functions are fixed quadratic structured functions. Denote a common complexity bound by $H$ . We first consider one such fixed term.

2. The permitted functions in the other two positions

This is the setup on D p.31.

Fix a dyadic subroot $K\subseteq J'$ and a finite family $\mathcal U$ of original intervals of use inside it. Also fix $D_h\ge2$ and $L\ge0$ . In position $e\in\{a,b\}$ , the sequence $z_{e,I}$ may be a sum of at most three terms:

z_{e,I}=\sum_{\ell=1}^{m_e}c_{e,\ell}V^\ell_{e,I}, \quad0\le m_e\le3,\quad |c_{e,\ell}|\le1. \tag{Allowed}

The coefficients $c_{e,\ell}$ are fixed, independent of $I$ and $s(I)$ . Each component must belong to one of the following three classes:

An original function $F_{e,s(I)}|_I$ ;
A current projection $P_{e,I}F_{e,s(I)}$ ;
A function supported on $I$ , denoted by $Z_{e,I}$ , satisfying

\sum_{I\in\mathcal U}|I|\|Z_{e,I}\|_{2,I}^2\le L^2|K|. \tag{Energy}

All original functions and projected functions in one position come from the same complete normalized blockwise sequence. Positions $a,b$ may use different blocks. Each projected component uses one fixed realization of an inherited projection family; the finitely many such families share the constant $D_h$ . Depths with no summation intervals are still filled in by inheritance and local reprojection.

Bilinearity reduces the combinations in the two positions to at most nine pairs of components. Replacing one position requires at most three opposing components. Once each class has been estimated, these fixed finite numbers of terms can be absorbed into the constants.

3. The Fourier supremum and size classes

Choose a fixed cutoff

\zeta\in C_c^\infty(\mathbb R),\quad 0\le\zeta\le1,\quad \zeta=1\text{ on }[-1/3,1/3],\quad \operatorname{supp}\zeta\subset(-1/2,1/2).

On the kernel support, $x,x+3t\in I$ , hence $|t|\le|I|/3$ , so inserting this cutoff leaves the form unchanged. Set $r=|I|$ and define

A_r=\sup_{\xi\in\mathbb R} \left|\int_{\mathbb R}Q(ru)\zeta(u)\mathrm e(-\xi u)\,\mathrm{d}u\right|.

Then $0\le A_r\le\|\zeta\|_1\le1$ and $\int\zeta\ge2/3$ .

Write

\mathcal H_I(\psi_a z_a,\psi_b z_b;Q) =|I|^{-2}\int w_I(x,t)Q(t) \psi_a(y_a)z_a(y_a)\psi_b(y_b)z_b(y_b) \,\mathrm{d}x\,\mathrm{d}t.

For an integer $m\ge0$ , let $\mathcal U_m$ be any chosen subfamily of intervals of use satisfying

\mathrm e^{-m-1}<A_{|I|}\le\mathrm e^{-m}

D Lemma 5.3 asserts that

\sum_{I\in\mathcal U_m}|I| \left|\mathcal H_I(\psi_a z_{a,I},\psi_b z_{b,I};Q)\right| \le \mathrm e^{-m} \operatorname{poly}\bigl(1+m,\log(2H),\log(2D_h),1+L\bigr)|K|. \tag{Band-sum}

Here $\operatorname{poly}$ denotes a polynomial bound of fixed finite degree. Neither its coefficients nor its degree may grow with the number of intervals. The key point is that the size of the projections in the complementary positions (the paper's high positions) enters only through $\log(2D_h)$ .

4. The scalar results used from H

The proof in D uses the scalar part of H Lemma 5.1 (pp.30–31). Set

\mathcal K=1+m+\log(2H).

The following facts concern the already fixed functions $\psi_a,\psi_b,Q$ and do not depend on the choice of functions in the other two positions.

First, the single-scale separation estimate in H Lemma 4.6 bounds the operator norm of the original bilinear form by $C A_r$ , with both functions measured in normalized $L^2(I)$ . On a small collection of exceptional scales, this permits direct estimation of the original form before expanding all the structured coefficients.

Second, H Lemmas 4.3–4.4 give scale-dependent expansions whose principal phases are

\mathrm e(\lambda_*t^2+\gamma t), \quad |\lambda_*|r^2\le D_1, \quad\log D_1\le\operatorname{poly}(\mathcal K).

The unscaled parameters $\lambda_*,\gamma$ come from combinations of bounded rational height of one fixed real parameter vector. The permitted combinations and scalar coefficients vary with the scale; the underlying parameter vector is not replaced by an unrelated one.

The truncation error satisfies more than a small single-scale bound. Each fixed phase pair $(\lambda_*,\gamma)$ has an absolute coefficient bound valid across scales and permitted decompositions. These bounds are summable over the fixed pairs, and their total can be made smaller than

\mathrm e^{-2m}(1+D_1)^{-M}

where the fixed integer $M\ge2$ is chosen larger than the degree of the later smooth-kernel polynomial cost plus one. Thus the tail is made smaller to absorb that cost. This is the pooled tail result from H used on D p.32, referred to below as the uniform coefficient tail bound across scales.

Finally, the height-grouping result in H Lemma 3.5 controls the number of scales with intermediate nonzero quadratic size $|\lambda_*|r^2$ or linear difference $r|\gamma-\gamma'|$ . That number is polynomial in $\mathcal K$ . On the remaining scales, the total errors from discarding very small quadratic terms and replacing nearby centers by representatives are also summable. H p.31 further gives a joint Bessel bound over the retained scales, intervals, and representative centers.

These are the scalar assumptions for the conditional argument in this part. We next check how varying functions enter the estimates.

5. Errors controlled by the $L^2$ size at each depth

Fix a depth $s$ . Intervals of use at that depth are pairwise disjoint. For original functions, normalization gives

\sum_{\substack{I\in\mathcal U\\s(I)=s}} \|F_{e,s}\|_{L^2(I)}^2\le|K|.

For projected functions, the contraction property of orthogonal projection on each interval gives the same bound. For an energy-type component, the global square sum in (Energy) gives the bound $L^2|K|$ at this depth.

There are at most three terms in (Allowed), so

\sum_{\substack{I\in\mathcal U\\s(I)=s}} \|z_{e,I}\|_{L^2(I)}^2\le C(1+L)^2|K|. \tag{One-depth}

Cauchy–Schwarz in the two positions then bounds the sum of the products of their ordinary $L^2$ norms by $C(1+L)^2|K|$ .

On exceptional scales, first apply the original single-scale bound $C A_r\le C\mathrm e^{-m}$ , then (One-depth). There are only polynomially many exceptional scales, so their contribution satisfies (Band-sum). This avoids a potentially large factor from the total absolute mass of the structured coefficients.

For fixed $\lambda_*$ , one can also calculate directly that

\sum_{\substack{r\text{ dyadic}\\|\lambda_*|r^2\le D_1}} |\lambda_*|r^2\le C D_1. \tag{Quadratic-sum}

If $\lambda_*=0$ , every term vanishes; otherwise the quantities decrease by a factor of $1/4$ at successive smaller scales. This handles the second term in

\mathrm e(\lambda_*t^2)=1+\bigl(\mathrm e(\lambda_*t^2)-1\bigr)

whose smooth-kernel norm contains the factor

|\lambda_*|r^2\operatorname{poly}(D_1).

Together with (One-depth) and the uniform coefficient tail bound, this makes the error summable over all scales. The geometric-series argument is applied to a fixed $\lambda_*$ throughout.

6. Why the mean-zero tensor expansion admits smooth-row estimates

For the linear-phase part, first absorb $\mathrm e(\gamma t)$ into two fixed multipliers, then express the unmodulated kernel $w_I$ in coordinates $(y_a,y_b)$ . The zero marginals and double primitive below concern this $w_I$ ; no such assertion is made directly for $w_I\mathrm e(\gamma t)$ . The absolute Jacobian is $|b-a|$ , fixed and nonzero. The four-direction cancellation condition includes directions $a,b$ , so integrating the transformed kernel in either variable with the other fixed gives zero.

Integrate once in each variable to obtain a double primitive. The zero marginal integrals ensure that it vanishes outside a rectangle strictly inside $I\times I$ . Expand this primitive in a smooth Fourier series on a slightly larger fixed rectangle, multiply by an interior cutoff equal to one on its support, and differentiate in both variables. This gives a tensor expansion of the original kernel.

Each one-variable function is the derivative of a compactly supported function and therefore has integral exactly zero. Fixed-order derivative bounds grow at most polynomially in the full Fourier index, while the coefficients decay faster than the reciprocal of any fixed polynomial. We may therefore prove the row estimate at one fixed full tensor index, then sum absolutely over the indices.

For a fixed position $e$ and frequency representative $\nu$ , the full multiplier is

m_\nu(y)=\mathrm e(\nu y)\psi_e(y), \quad \|m_\nu\|_\infty\le1.

The function $\psi_e$ is fixed in advance on the common initial root. A given $\nu$ uses the same $m_\nu$ on every interval. The corresponding row is

W_{I,\nu}f=|I|^{-1/2}\int_I m_\nu(y)\omega_{I,\nu}(y)f(y)\,\mathrm{d}y,

where $\omega_{I,\nu}$ is the smooth mean-zero function just constructed.

7. Estimates for original, projected, and energy-type functions

7.1 Original function sequences

Fix a tensor index and the set of interval–frequency representatives selected in the source proof. For this actual selected set, H p.31 gives an ordinary joint Bessel bound with operator norm $B_H$ . Its square is polynomial in $\mathcal K$ and depends polynomially on the current tensor index. Denote the bounds for the smooth function and its scaled first derivative by $A_{\mathrm{bump}}$ .

The fixed multiplier family $\mathcal N$ satisfies

\log(2+\#\mathcal N)\le\operatorname{poly}(\mathcal K).

This follows by counting the dimension of the fixed parameter vector, the rational heights, and the uniform truncation. The frequencies themselves need not be bounded.

Retain only rows supported on the subroot $K$ . Extending any $L^2(K)$ test function by zero to the common initial root preserves the original joint Bessel bound. D Lemma 4.6 from the preceding part therefore gives

\left(\sum_{(I,\nu)} |W_{I,\nu}f_{e,s(I)}|^2\right)^{1/2} \le C(B_H+A_{\mathrm{bump}}) \bigl(\log(2d_{\mathrm{car}})\bigr)^2|K|^{1/2}, \quad d_{\mathrm{car}}=2+\#\mathcal N. \tag{Raw-rows}

The right-hand side still has only the required polynomial loss.

For a fixed linear phase pair in the uniform tail bound, fix its frequencies and full multipliers before estimating its rows. There is then only one multiplier index. Sum over the absolute coefficient bounds for phase pairs only afterward.

7.2 Current projected function sequences

The original function in (Raw-rows) cannot simply be replaced by its interval-dependent projection. A new ordinary joint operator bound is needed first.

Applied to an inherited projection family, the Compression statement in H Lemma 3.4 (p.9 onward) asserts the following. If the original rows supported on decision intervals $I$ have joint Bessel norm $B_H$ , then the composed rows

A_{I,\nu}=W_{I,\nu}P_{e,I}

have ordinary joint operator norm

B_H\,\mathfrak C(1,D_h), \quad\mathfrak C(1,D_h)\le C\bigl(\log(2D_h)\bigr)^C. \tag{Compression-one}

This statement applies to arbitrary fixed functions and is another explicit result from H used here. The rows and projection family must first be fixed by the actual construction.

View the original rows as rows on the augmented space, ignoring its private coordinates. Their Bessel norm is unchanged. The adjoint of a composed row is

A_{I,\nu}^*=P_{e,I}W_{I,\nu}^*.

Its output on the whole of $I$ is a constant-coefficient combination of fixed available column functions, and its input depends only on $I$ . Order the support depths increasingly, place all rows at one depth in one round, use the full partition at that depth for both support and observation, and use $F_{e,s}$ in each round. The refinement relation and nondecreasing indices satisfy D Lemma 4.4.

Applying (Compression-one) and then the varying-function row estimate gives the source equation (5.14):

\left(\sum_{(I,\nu)} |W_{I,\nu}P_{e,I}F_{e,s(I)}|^2\right)^{1/2} \le C B_H\mathfrak C(1,D_h)\log(2D_h)|K|^{1/2}. \tag{Projected-rows}

No factor grows with the total depth.

7.3 Energy-type functions

For $Z_{e,I}$ , use the Bessel bound over separated frequency centers on each fixed interval from the same row construction in H, then sum over intervals. Equation (Energy) gives

\sum_{I,\nu}|W_{I,\nu}Z_{e,I}|^2 \le \operatorname{poly}(\mathcal K)\,L^2|K|.

This requires neither the projection-composition result nor an identification of the different $Z_{e,I}$ as one function.

8. Small retained frequency coefficients and summation over scales

The Fourier-supremum test in H shows that the combined scalar coefficient of each cluster of linear centers is $O(\mathrm e^{-m})$ . Test the cluster's representative frequency against the original $Q(r\,\cdot)$ . Its own coefficient is multiplied by the fixed nonzero number $\int\zeta$ ; separation makes the other clusters contribute controlled Fourier tails. The earlier quadratic, truncation, and nearby-center replacement errors are already bounded at the same scale.

This bound applies after combining the coefficients within a cluster. It does not give a small bound for every original large coefficient after taking absolute values separately.

At a fixed tensor index, $|I|$ times the bilinear form is a product of two normalized row values, multiplied by the fixed Jacobian factor and the cluster coefficient. Apply Cauchy–Schwarz over all indices of the paired rows, then (Raw-rows), (Projected-rows), or the energy bound. The result is

\mathrm e^{-m}\operatorname{poly} \bigl(\mathcal K,\log(2D_h),1+L\bigr)|K|.

The rapid decay of the tensor coefficients absorbs the polynomial dependence of derivative bounds and Bessel constants on the tensor index. Adding the exceptional scales and tails controlled in Section 5 proves (Band-sum).

If $A_r=0$ , the compactly supported $L^1$ function $u\mapsto Q(ru)\zeta(u)$ has identically zero Fourier transform and hence vanishes almost everywhere. On the relevant region $|u|\le1/3$ , we have $\zeta=1$ , so the corresponding bilinear form vanishes. Every positive value of $A_r$ belongs to one of the stated size classes.

9. The full statement for a fixed predictor family

This is D Corollary 5.4, p.34, equations (5.15)–(5.17).

Fix the functions $\phi_i,\phi_j$ in the fixed-atom positions with complexity at most $H_{\mathrm{low}}$ , and fix a precision parameter $B\ge1$ . There exist a finite index set $\mathcal A$ , a constant $D_{\mathrm{pred}}$ , fixed triples

(\psi_{a,\alpha},\psi_{b,\alpha},\lambda_\alpha) \quad(\alpha\in\mathcal A),

and scalar coefficients depending only on the dyadic scale $r$ , denoted by $c_{r,\alpha}$ , such that

\|\psi_{e,\alpha}\|_\infty\le1, \quad \#\mathcal A,\ \operatorname{complexity}(\psi_{e,\alpha}),\ |c_{r,\alpha}|\le D_{\mathrm{pred}},

\log D_{\mathrm{pred}} \le\operatorname{poly}\bigl(1+B+\log(2H_{\mathrm{low}})\bigr).

If $|\lambda_\alpha|r^2>D_{\mathrm{pred}}$ , then $c_{r,\alpha}=0$ .

The triples and all scale coefficients depend only on the fixed functions in the fixed-atom positions, their scalar construction, and $B$ . They must be fixed before choosing the functions in the other two positions and their final projection families. Define

\mathcal T_I^\alpha(z_a,z_b) =|I|^{-2}\int w_I(x,t)\mathrm e(\lambda_\alpha t^2) \psi_{a,\alpha}(y_a)z_a(y_a) \psi_{b,\alpha}(y_b)z_b(y_b) \,\mathrm{d}x\,\mathrm{d}t.

For every dyadic $K\subseteq J'$ , every finite family of intervals of use, and every pair satisfying (Allowed), define the actual error $e_I$ by

\mathcal L_I^{\phi_i,\phi_j}(z_{a,I},z_{b,I}) =\sum_{\alpha\in\mathcal A}c_{|I|,\alpha} \mathcal T_I^\alpha(z_{a,I},z_{b,I})+e_I

Then

\sum_{I\in\mathcal U}|I|\,|e_I| \le\mathrm e^{-B} \operatorname{poly}\bigl(\log(2D_h),1+L\bigr)|K|. \tag{Predictor-error}

The error $e_I$ may depend on the current functions in the two positions. The triples and coefficients remain fixed; the conclusion bounds the sum of the actual errors for every such choice.

10. Why the truncation does not depend on later projection sizes

The separation expansion in H Lemma 4.5 has rapidly decaying total coefficient mass in successive shells, with prescribed polynomial growth of shell complexity. The polynomial in (Band-sum) can be enlarged to a product of two nonnegative polynomials:

P_1(1+m,\log(2H))\,P_2(\log(2D_h),1+L).

Thus the separation shells can first be truncated using only $B,H_{\mathrm{low}}$ , with sufficiently small total error from the discarded high shells. For each retained term, discard sufficiently small values of $A_r$ , equivalently sufficiently large $m$ . The geometric decay $\mathrm e^{-m}$ absorbs the fixed polynomial growth of $P_1$ , while $P_2$ remains on the right-hand side of the final error estimate.

For the finitely many remaining shells and size classes, truncate again using the uniform coefficient tail bound from Section 4. The logarithms of these thresholds are bounded by a fixed power of $1+B+\log(2H_{\mathrm{low}})$ . The unscaled phase parameters come from finitely many choices of bounded rational height, so the retained triples form a fixed finite family.

Split the linear phase between the two fixed multipliers using

\mathrm e(\gamma t) =\mathrm e\!\left(-\frac{\gamma y_a}{b-a}\right) \mathrm e\!\left(\frac{\gamma y_b}{b-a}\right)

Move scalar magnitudes into the coefficients so that both multipliers retain absolute value at most one. Their real oscillation parameters may be large, but those parameters do not count toward complexity in the original definition of a structured function. This gives the triples, coefficients, and complexity bounds in Section 9.

11. Why the same family works on every subroot

Once the scalar truncations, phases, and multipliers have been chosen, they remain unchanged on any $K\subseteq J'$ . Retain only the source rows supported on intervals in the current family of use.

Extending any $L^2(K)$ function by zero to $J'$ preserves the ordinary Bessel bound. Restricting the original sequence to $K$ preserves normalization. Locally reproject the projection family on $K$ using the retained columns; the column-function constants from the preceding part do not increase. The estimates in Sections 5–7 therefore apply again on $K$ , with a size factor $|K|^{1/2}$ for each position.

Summing the same discarded scalar shells, size classes, and uniform tails gives (Predictor-error). The argument reapplies operator estimates on the subroot; it does not distribute an arbitrary parent-root error in proportion to subroot length.

Source coverage and the next step

D pp.30–31: fixed structured functions in the fixed-atom positions, the three permitted classes of opposing functions, full multipliers, and the Fourier supremum.
D pp.32–34, Lemma 5.3: exceptional scales, the uniform coefficient tail bound, and varying-function estimates for smooth rows and rows composed with projections.
D p.34, Corollary 5.4: fixing the predictor family before the other two positions and controlling the actual error on every subroot.
H Lemmas 3.4–3.5, 4.3–4.6, and the proof of Lemma 5.1: the fixed-function and scalar assumptions used explicitly here. Their proofs have not all been independently verified.

The next part, on D Lemma 5.5, places these fixed predictors in the augmented space and uses their private coordinates to control moments of delayed projection residuals. The required replacement is $P_{s+K_0}F_s$ , rather than $P_{s+K_0}F_{s+K_0}$ . The predictor-family estimate alone does not establish that replacement.