A Mathematical Theory of Near-Field Super-Resolution

1.1. What can a finite aperture distinguish?

• Far-field responses depend only on angle and are blind to range along a fixed ray, whereas Fresnel wavefront curvature makes the response range-dependent and couples range with angle.The Fresnel interaction is governed by linear and quadratic aperture-phase coordinates rather than a translation-invariant range-angle kernel.Source: §1.1 · p. 2, §1.2 · p. 3

1.3. Why angular separation is not the right invariant.

• Angular separation alone does not control Fresnel interactions because cancellation depends jointly on linear and quadratic phase across the finite aperture, while tangent and derivative channels depend on the individual points’ parameterization.The resulting interaction bounds supply the interpolation, curvature, and leakage estimates required by the dual certificate.Source: §1.3 · p. 5

1.4. Quadratic-phase aperture certification.

• QPAC certifies sufficient cancellation in finite quadratic-phase sums across the kernel, normalized tangent, and higher angular derivative channels, without imposing a universal scalar separation threshold.Its derivative, lag-correlation, and residue-linear branches address nonstationary phases, constructive alignment, and near-rational curvature, respectively.Source: §1.4 · p. 5, §1.9 · p. 9

1.5. Dual certificates: interpolation, curvature, leakage.

• A gauged Hermite dual certificate interpolates source values, has negative angular curvature near support points, and uses QPAC to control off-support leakage.The range coordinate is discrete, leaving angle as the only continuous local direction.Source: §1.5 · p. 6

1.6. Main recovery result.

• Under support-uniform QPAC estimates, every sparse measure in the prescribed support class uniquely minimizes the noiseless semi-discrete Fresnel TV problem, uniformly over nonzero complex amplitudes.This is a sufficient criterion; the paper claims no converse.Source: §1.6 · p. 6, §1.9 · p. 9

1.7. Computation and the finite-harmonic lift.

• The finite-harmonic Bessel-Vandermonde lift is a computational approximation with an explicit uniform truncation error and an SDP formulation, not an additional exact-recovery theorem.The exact theorem concerns the semi-discrete Fresnel TV problem without the lift.Source: §1.7 · p. 6, §1.9 · p. 9

1.8. Comparison with related work and sparse recovery.

• For the fixed deterministic Fresnel aperture, the contribution is an explicit finite-aperture arithmetic route to controlling dual-certificate channels, complementary to general geometric theories for non-translation-invariant recovery.The paper presents QPAC as an operator-specific sufficient criterion, not as a claim that broader geometric frameworks do not exist.Source: §1.8 · pp. 7–8

1.9. Contributions and organization.

• The paper’s deterministic bounds cover finite quadratic sums with channel-dependent coefficients and support-uniform envelopes for the Fresnel kernel, normalized tangent, and higher angular derivatives.These bounds feed the support-interpolation, near-support decay, and far-leakage budgets in the recovery condition.Source: §1.9 · p. 9
• A^† denotes the Moore-Penrose pseudoinverse, ⊙ entrywise multiplication, B2 the Euclidean unit ball, and ρsp(G) the spectral radius.Complex inner products are conjugate-linear in their first argument.Source: §1.9 · p. 10
• The fixed certificate taper is denoted ρ_n, with b_n defined as ρ_n divided by the sum of the taper weights.Source: §1.9 · p. 10
• Channel-dependent scalar sequences in the QPAC estimates are not extra tapers; when ρ_n vanishes, weighted Hilbert-space interpretation is restricted to active aperture indices, while atom independence is in C^N_r.The active aperture is A_ρ = {n : ρ_n > 0}.Source: §1.9 · p. 10
• For matrices, the Frobenius inner product is trace(X^H Y), A^H denotes conjugate transpose, T* is the operator adjoint, and Hermitian matrices satisfy R = R^H.Source: §1.9 · p. 10

2.1. The semi-discrete Fresnel model.

• The semi-discrete model uses a finite grid of ranges, continuous angles, complex measures, and TV minimization under Fresnel measurements.The feasible domain is Q = {1, . . . , Nd} × Θ.Source: §2.1 · p. 11

2.2. Quadratic phase interactions.

• Each ordered interaction has a finite-aperture phase with angular mismatch ω1 and curvature mismatch ω2; the far-field limit sets ω2 = 0.Source: §2.2 · p. 12

2.3. Support classes and near-far decomposition.

• Support-uniform QPAC envelopes bound support interactions using derivative, lag-correlation, or residue-linear cancellation, with ℓ1 and Cauchy-Schwarz as fallback bounds.The certifying branch may vary by ordered pair and channel.Source: §2.3 · pp. 13–14

2.4. The QPAC recovery number.

• QPAC combines support-interpolation, near-support curvature, and far-region leakage budgets to control the strict TV dual certificate.A near-support budget below one implies strict certificate decay on the punctured feasible near set.Source: §2.4 · pp. 14, 16

2.5. Main exact recovery theorem.

• Derivative-separation and residue-lag cancellation support classes are two specializations of the same recovery theorem, with lag classes controlled by complete residue lag-phase nonalignment.The derivative class also bounds a curvature factor; that curvature bound is not itself a separation condition.Source: §2.5 · pp. 18–19
• The exact theorem assumes ranges on a finite admissible grid and continuous angles; fully continuous range-angle recovery would require a two-parameter certificate.The one-dimensional angular Hermite certificate suffices for this semi-discrete model.Source: §2.5 · p. 19
• If a prescribed support subclass satisfies the QPAC threshold, the main theorem gives exact recovery uniformly over its supports, independently of nonzero amplitudes and their phases.After support recovery, amplitudes are obtained by finite-dimensional least squares.Source: §2.5 · pp. 19–20

2.6. Finite-harmonic lifted Fresnel localization.

• The finite-harmonic Bessel-Vandermonde lift is a computational surrogate, not a second exact-recovery theorem, and uniformly approximates a fixed QPAC certificate with truncation margin tending to zero as harmonic orders increase.The lifted SDP produces its own dual polynomial, characterized separately by Proposition 4.1.Source: §2.5 · p. 20

2.7. Far-field limit.

• In the far-field limit, range disappears from the narrowband response, leaving a linear-phase finite-aperture certificate geometry in angular spatial frequency κ(θ).Same-bearing sources at different ranges then have identical steering vectors and are recoverable only through their combined complex amplitude.Source: §2.7 · p. 20

3.1. Quadratic exponential sums and finite-aperture cancellation.

• QPAC bounds incomplete quadratic sums with channel-dependent coefficients over finite apertures, producing explicit support-uniform envelopes for the Fresnel certificate.The estimates are designed for physical support/evaluation cells rather than a single fixed phase pair or an asymptotic large-aperture regime.Source: §3.1 · pp. 21–22, §3 · p. 21
• Three complementary QPAC branches control nonstationary phase, lag-correlation alignment, and near-rational curvature through residue-class linear estimates.The lag-correlation branch retains complete lag-pair phases and bounds real lag terms with interval cosine majorants.Source: §3.1 · p. 22
• The resulting quadratic-sum budgets certify Hermite interpolation invertibility, strict near-support angular decay, and far-region leakage control.These are the certificate tasks that turn finite-aperture cancellation into a deterministic recovery principle.Source: §3.1 · pp. 22–23

3.2. Scalar QPAC branches.

• Scalar QPAC estimates are combined from universal and analytic cancellation bounds, with separate branches for nonstationary phases and near-rational curvature.The pointwise residue split is distinct from the support-uniform lag-correlation construction.Source: §3.2 · pp. 23–24

3.3. From scalar sums to certificate envelopes.

• Gauged Hermite channels K, H, dK, and dH control interpolation and tangent interactions, while higher angular derivatives bound local curvature and Taylor remainders.The scalar pairwise bounds are converted to physical-cell envelopes using phase intervals, tangent-arc hulls, and derivative-profile bounds.Source: §3 · p. 21, §3.3 · p. 24

3.4. Physical domain, taper, and tangent normalization.

• The paper normalizes the taper and tangent profiles to construct gauged Fresnel atoms whose angular derivative is orthogonal to each atom.Ordered interactions then have a quadratic phase with linear and curvature mismatch coordinates, and their normalized pairwise kernels form the four Hermite channels.Source: §3.5 · pp. 25–26, §3.4 · p. 25

3.6. Higher angular derivative coefficient families.

• Second- and third-order evaluation-side angular derivative channels are built from channel-specific coefficient families to control variation of the gauged kernel.Their derivative-size fallback bounds need not be capped by one, unlike the normalized channels.Source: §3.2 · p. 24, §3.6 · pp. 26–27

3.7. Pairwise channel bounds.

• A fixed-pair QPAC bound is pointwise, whereas recovery requires certified suprema over continua of support/evaluation configurations.The constants certify support-support interactions, near-set derivative sums, and far-region kernel sums; each source contribution is evaluated at a common point before taking the supremum.Source: §3.7 · p. 27, §3.8 · p. 27

4.1. Finite-harmonic Fresnel representation.

• A truncated Bessel-Vandermonde expansion approximates Fresnel atoms uniformly, with an explicit truncation error bound.The construction discretizes range on a finite grid while retaining continuous angles.Source: §4 · p. 28, §4.1 · p. 29

4.2. Lifted atomic norm and finite Fresnel operator.

• The finite-harmonic atoms define a lifted atomic norm and operator, with a norm assumption satisfied when the angular set contains a nondegenerate interval.The lifted model represents semi-discrete measures using a finite-dimensional matrix variable.Source: §4.2 · p. 29

4.3. Lifted dual polynomial and SDP feasibility.

• The lifted dual uses bounded trigonometric polynomials with a semidefinite representation; imposing the modulus bound on the full circle is conservative for the physical angular interval.Restricted angular arcs can instead use localizing Toeplitz constraints.Source: §4.3 · p. 30

4.4. range-angle extraction from lifted dual peaks.

• Near-unit peaks of the lifted dual polynomial provide candidate range-angle locations, but saturation identifies optimizer atoms rather than guaranteeing the ground-truth support.The range row gives the range estimate and the angular peak gives the angle estimate; near-saturation accommodates solver accuracy and numerical peak localization.Source: §4.4 · pp. 31–32

4.5. Connection with the Fresnel QPAC certificate.

• For a fixed QPAC dual vector, the finite-harmonic polynomial approximates the ideal Fresnel certificate uniformly, and sufficiently small truncation error preserves its separation from off-neighborhood points.The threshold interval is nonempty when the approximation error is less than half the QPAC margin.Source: §4.5 · p. 33

4. Finite-harmonic lifting for Fresnel dual localization.

• The lifted SDP is a computational surrogate, not a second exact-recovery theorem, and its computed dual polynomial need not equal the polynomial generated by the QPAC vector.Dual saturation alone does not assert equality with the ground-truth support.Source: §4.5 · p. 33

5. Numerical QPAC evaluations and computational illustrations.

• The derivative-route support class has two independently varying angles, one source on each of two range rows, while the correlation-route class fixes a common bearing across two ranges.The examples evaluate distinct QPAC branches and the support-interpolation, near-support, and far-region budgets.Source: §5 · p. 33, §5.1 · p. 34, §5.2 · p. 34

5.1. Derivative-route QPAC example.

• The derivative-route calculation gives a QPAC recovery number below one for the prescribed support class.The two angles vary independently across two range rows; the near and far budgets cover the continuous class using analytic padding between grid points.Source: §5.1 · p. 34

5.2. Correlation-route QPAC example.

• For two sources at a common bearing and different ranges, the correlation-route example reports CQPAC ≈ 0.825990 < 1.The derivative threshold is not met, so support interpolation instead uses the lag-correlation bound.Source: §5.2 · pp. 34–35

5.3. Lifted dual-polynomial localization.

• The finite-harmonic lift uses 73 harmonics across eight logarithmically spaced range bins spanning a factor of 12 in range.The selected Jacobi-Anger orders keep the finite-harmonic approximation error small.Source: §5.3 · p. 36
• The lifted dual polynomial localizes two sources with maximum paired angular error 4.77796 × 10^-4 rad and exact recovery of both semi-discrete range bins.Its near-unit peaks identify the active range bins and continuous angles; the finite-harmonic approximation uses a truncation radius exceeding the realized Fresnel-to-lifted data mismatch.Source: §5.3 · p. 36, Figure 3 · p. 37

5.4. QPAC channel envelopes and higher-derivative control.

• Cellwise support-uniform envelopes add conservatism to pointwise QPAC bounds, and their channels feed the recovery theorem’s three budgets.K, H, dK, and dH control Hermite interpolation; second derivatives control local curvature, and third derivatives control the Taylor remainder.Source: §5.3 · p. 37, §5.4 · p. 38

6. Discussion

• Near-field interactions follow a finite quadratic phase path, so angular separation alone does not determine coherence.Angular and curvature coordinates interact over the finite aperture; coherence depends on whether the phase adds coherently or cancels.Source: §6 · pp. 38–39
• QPAC is a deterministic sufficient certificate, not a necessary characterization or a universal scalar minimum-separation threshold.Its bounds control the kernel, normalized angular tangent channels, and angular derivative channels used by the gauged Hermite certificate.Source: §6 · pp. 39–41
• The theorem restricts range to a finite admissible set and leaves angle continuous; fully continuous range-angle recovery would require a two-parameter certificate.That extension would also need range, angle, and mixed derivative control.Source: §6 · p. 41
• The finite-harmonic lift is a computable surrogate with explicit truncation control, but its perturbation result does not identify the QPAC dual vector with the lifted SDP optimizer.For physical spherical-wave data, paraxial mismatch is a separate approximation; exact physical recovery would require an asymptotic regime or a spherical-wave certificate theory.Source: §6 · p. 41
• The authors identify sharper envelopes, continuous-range certificates, and QPAC bounds for more general apertures as future directions.Planar, volumetric, sparse, and conformal arrays require multidimensional quadratic-sum envelopes and corresponding multi-parameter derivative channels.Source: §6 · pp. 41–42
• In the far-field limit the quadratic coordinate disappears, range becomes unobservable from single-carrier spatial aperture data, and the theory reduces to Fourier-type angular geometry.The work’s reproducibility archive contains scripts for the QPAC examples, lifted localization, and channel-bound plots.Source: §6 · p. 42

A.3. The adjoint family and the TV dual problem.

• The formal dual uses the adjoint image of a dual vector as a range-indexed trigonometric family, with modulus bounded by one.The proof of uniqueness needs weak duality and one strict certificate, rather than the full formal-dual derivation.Source: §A.1 · p. 45, §A.3 · p. 46

A.4. TV dual certificate criterion.

• A dual family that interpolates source signs, stays strictly below unit modulus off support, and has independent active atoms certifies unique TV recovery.Feasibility makes every competitor’s pairing with the dual family equal to the true measure’s pairing; strictness confines minimizers to the support.Source: §A.1 · p. 45, §A.4 · pp. 46–47

A.5. Gauged Fresnel atoms.

• The gauge removes a local phase so the value atom is orthogonal to its angular tangent, and the resulting normalized atoms have unit norm.Value and tangent directions are then used to form the gauged kernels.Source: §A.1 · p. 45, §A.5 · p. 48

A.2. Vector measures and the TV program.

• The TV program minimizes total variation over semi-discrete measures subject to matching the noiseless Fresnel data.The unknown is a measure over finite range indices and continuous angles.Source: §A.1 · p. 45, §A.2 · p. 45

A.6. Gauged value and tangent kernels.

• The gauged value and tangent kernels support Hermite interpolation of the source signs with zero angular derivative at each support point.The pre-certificate is a linear combination of value and tangent kernels.Source: §A.6 · p. 49, §A.7 · p. 49

A.7. The gauged Hermite pre-certificate.

• The gauged Hermite construction imposes value interpolation and zero angular derivative at every support point.Its interpolation matrix combines the value and tangent kernel evaluations into a block system.Source: §A.6 · p. 49, §A.7 · p. 49

A.8. Block Neumann domination.

• If the off-diagonal Hermite interactions are dominated by a comparison matrix with spectral radius below one, a Neumann argument gives invertibility and coefficient bounds.The comparison reduces the interactions to a nonnegative 2 × 2 system.Source: §A.8 · pp. 50–51

A.9. From the gauged certificate to an adjoint TV certificate.

• The gauged certificate is stationary at support points, but the ungauged adjoint certificate need not be; stationarity is used only to prove local modulus decay.The TV criterion instead requires sign interpolation and strict off-support boundedness for the ungauged certificate.Source: §A.9 · p. 53

A.10. Atom linear independence.

• Invertibility of the gauged Hermite matrix rules out any nontrivial dependence among active Fresnel atoms.Thus atom independence required by the TV dual-certificate criterion follows from Hermite invertibility rather than serving as an additional main-theorem hypothesis.Source: §A.10 · p. 53

B.1. Scalar notation and finite differences.

• B.1 establishes the finite-difference and phase-separation notation used to state the scalar QPAC bounds.The fourth-order Abel summation argument includes an auxiliary endpoint.Source: §B.1 · pp. 54–55

B.2. The scalar QPAC best-branch estimate.

• The scalar QPAC estimate takes the minimum of a universal cap and three cancellation bounds, with the derivative branch requiring fourth-order flat ends.For N ≥ 9 and Qmax in {2, …, N}, the flat-end condition is needed only for the derivative branch; the other branches hold for arbitrary finite coefficient sequences, and gauged-kernel coefficients inherit flat ends from the taper.Source: §B.1 · p. 55, §B.2 · p. 55

B.3. Derivative branch.

• Fourth-order Abel summation bounds the derivative branch when discrete phase increments stay separated from 2πZ and flat ends eliminate boundary terms.The proof controls the recursively differentiated coefficient sequence using finite differences of the coefficients and reciprocal phase increments.Source: §B.3 · p. 57

B.4. Pointwise residue splitting.

• Residue splitting gives an exact decomposition before absolute values, and minimizing the sum of residue-class magnitudes over Q = 2,…,Qmax yields a pointwise bound.Q = 1 is excluded because its single residue class gives only the original sum and no independent cancellation estimate.Source: §B.4 · p. 60

B.5. Residue-linear branch.

• Splitting the aperture into residue classes near a rational quadratic phase yields a residue-linear bound by applying a first-order Abel estimate to each class.The bound is minimized over the admissible rational approximants; a vanishing Abel denominator uses the trivial estimate for that class.Source: §B.5 · pp. 61–62

B.7. Pairwise reductions for K, H, dK, dH.

• The gauged Fresnel atom uses a normalized, constant-phase-adjusted Fresnel atom, and all four pairwise channels K, H, dK, and dH reduce to weighted scalar quadratic sums with a shared ordered phase.Their coefficient sequences inherit the taper's fourth-order flat-end condition; the support or evaluation tangent supplies the amplitude factor according to the channel.Source: §B.7 · pp. 63–64, §B.6 · p. 63

B.8. First angular derivatives.

• Differentiating with respect to the evaluation angle identifies the physical first derivatives of K and H with the normalized evaluation-side tangent channels dK and dH, scaled by the tangent norm.The support point remains fixed in these derivatives.Source: §B.8 · pp. 64–65

B.9. Higher angular derivative reductions.

• Second and third evaluation-angle derivatives add explicit aperture multipliers but preserve the underlying ordered quadratic phase, so they also admit scalar quadratic-sum representations.The derivative multipliers are eΠ2 and eΠ3.Source: §B.9 · p. 65

B.10. Dictionary form of eΠ2 and eΠ3.

• Writing eΠ2 and eΠ3 as finite dictionaries of fixed aperture monomials enables certified support-uniform bounds, but does not give the exact supremum over the nonlinear higher-derivative family.The construction gives an implementable upper bound from finite atoms and scalar QPAC branch envelopes.Source: §B.10 · pp. 66–67

B.11. Physical cells and induced phase boxes.

• Physical support/evaluation cells induce intervals for the phase increments and tangent parameters; tangent intervals are formed from endpoint hulls because cos θ can be negative.Source: §B.10 · p. 67, §B.11 · p. 67

B.12. Cellwise branch admissibility.

• Cellwise branch admissibility requires branch conditions to hold uniformly over the induced phase box; the LCS branch uses certified real lag bounds, while residue-linear and derivative branches require their respective phase or denominator conditions.The LCS branch has no denominator condition.Source: §B.11 · p. 67, §B.12 · pp. 67–69

B.13. Cellwise coefficient envelopes.

• Cellwise coefficient envelopes are defined to bound each channel throughout a physical cell, with derivative-branch bounds and lag-correlation support envelopes constructed for the eight listed channels.The listed channels are K, H, dK, dH, d2K, d3K, d2H, and d3H.Source: §B.13 · p. 69
• Certified cellwise envelopes combine lag-based minima over Q with residue-linear branches over admissible (q, A), while a trivial branch uses an ℓ1 coefficient bound.The residue-linear branch is valid on the cell when certified bounds hold for each residue class; an empty admissible family has minimum +∞.Source: §B.13 · pp. 69–71
• Because scalar QPAC estimates do not encode Hermite-channel normalization, unit weighted norms and Cauchy–Schwarz separately give a universal unit bound for four normalized channels.The bound holds independently of oscillatory cancellation.Source: §B.14 · p. 71

B.14. Cauchy-Schwarz bounds.

• For every pair (u, v), |K_uv|, |H_uv|, |dK_uv|, and |dH_uv| are at most 1.The bound follows from Cauchy–Schwarz applied to inner products of unit-norm gauged value and tangent atoms.Source: §B.14 · p. 71
• The second and third angular derivative channels are not normalized inner products, so their Cauchy–Schwarz caps instead depend on weighted ℓ2 sizes of derivative multipliers.The derivative-size caps apply to d_mK and d_mH for m = 2, 3.Source: §B.14 · pp. 71–72

B.15. Cellwise support-uniform QPAC theorem.

• On each compact physical cell, the channel bound is the minimum of valid whole-cell candidates, omitting inadmissible branches and using derivative-size caps for higher derivatives.The candidates include the trivial ℓ1 cap and appropriate Cauchy–Schwarz caps.Source: §B.15 · p. 72

Appendix C. Near-curvature estimates

• A positive near-curvature margin and a QPAC near number below one yield strict certificate-modulus decay on punctured feasible near intervals.The argument uses Hermite-system coefficient bounds, support tangent-energy lower bounds, and local derivative bounds through order three.Source: Appendix C · p. 73

C.1. Local notation.

• The gauged Hermite conditions interpolate prescribed phases at support points while setting the angular derivative to zero there.The local proof studies whether the certificate modulus is strictly below one away from each support point.Source: §C.1 · p. 74

C.2. Tangent-energy lower bound.

• The gauge defines tangent energy from weighted, centered aperture moments and assumes that energy remains positive across the admissible interval.The stated gauge makes the angular derivative proportional to the centered first- and second-order aperture coordinates.Source: §C.2 · p. 74

C.3. Coefficient bounds from the Hermite system.

• The Hermite-system estimates bound the interpolation-coefficient corrections by ΞK for αj−evj and ΞH for βj, uniformly over the support class and gauged phases.These are the coefficient estimates used in the near-curvature argument.Source: §C.3 · p. 75

C.4. Local derivative bounds.

• Local channel envelopes are certified on complete near sets, and the gauged atom’s normalized self-correlation has real second derivative equal to negative weighted tangent energy.The Taylor argument uses local evaluation-angle derivative bounds.Source: §C.4 · p. 75, §C.5 · p. 76

C.5. Curvature at the support.

• The retained curvature margin is mnear = 2σ2 − Ecurv, where Ecurv collects coefficient-correction and off-diagonal interaction losses from ideal diagonal curvature.The curvature lemma bounds the certificate’s second derivative using this margin.Source: §C.5 · pp. 76–77

C.6. Taylor proof of local strict decay.

• When mnear > 0 and ηnear(ϖloc) < 1, Taylor’s bound gives certificate modulus strictly below one at every nonzero feasible local offset.The proof uses Hermite interpolation and an integral remainder involving the third derivative.Source: §C.6 · pp. 77–78
• Distinct same-range support points cannot lie within ϖloc, because interpolation requires modulus one there while local strict decay requires modulus below one.The contradiction applies under the hypotheses of the strict-decay theorem.Source: §C.6 · p. 78

C.7. Support-uniform near set statement.

• The strict bound holds throughout the punctured feasible near set; its derivative estimates are certified on the corresponding complete near sets.The off-support set is Q \ S.Source: §C.7 · p. 79

Appendix D. Proof of the support-uniform Fresnel theorem

• The appendix assembles Hermite-system bounds, QPAC kernel estimates, near-support strict decay, and TV duality to prove the main theorem.Source: Appendix D · p. 79

D.1. Statement proved in this appendix.

• The prescribed support class satisfies pairwise support-interaction bounds, and the support-uniform QPAC budgets are each below one.These conditions supply the estimates used in the appendix's exact-recovery proof.Source: §D.1 · p. 79

D.2. Raw atoms, adjoint polynomials, and TV duality.

• The proof applies TV duality directly to raw Fresnel atoms by constructing an adjoint family through a gauged Hermite certificate and then removing the gauge.The raw atoms and their adjoint polynomials provide the certificate for the semi-discrete Fresnel problem.Source: §D.2 · pp. 79–80

D.3. Hermite interpolation and coefficient bounds.

• Gauged Hermite interpolation matches the prescribed phases and sets angular derivatives to zero; support-support bounds ensure a unique system solution with bounded coefficient corrections.The active raw Fresnel atoms are also linearly independent.Source: §D.3 · p. 80

D.4. Strict dual feasibility on the far set.

• On the far set, the certificate has magnitude strictly below one; if that set is empty, the claim is vacuous and its budget is zero.Source: §D.4 · p. 80

• The domain is divided into the support, a near neighborhood, and the far set, so certificate feasibility is checked region by region.Source: §D.5 · p. 80
• The gauged certificate interpolates unit-modulus phases on the support and is strictly bounded by one on the punctured near and far sets.The near-set and far-set bounds are supplied separately before combining them globally.Source: §D.5 · pp. 80–81
• Removing the unimodular gauge gives a raw Fresnel adjoint polynomial matching the target signs and strictly below one off support.Source: §D.5 · p. 81

• The active raw Fresnel atoms are linearly independent, completing the remaining hypotheses of the dual-certificate proposition.Source: §D.6 · p. 81
• The optimization problem minimizes total variation among measures with the same Fresnel measurements as the target.Source: §D.6 · p. 81
• The support-uniform Fresnel exact-recovery theorem is thereby proved.Source: §D.6 · p. 81

Appendix E. Finite-harmonic Bessel-Vandermonde expansions

• The appendix's lift is a finite-harmonic approximation of the semi-discrete Fresnel atom, not an additional exact-recovery theorem.Source: Appendix E · p. 82

E.1. Aperture and atom conventions.

• The aperture and atom setup specifies admissible ranges and angles in a non-occluding near-field regime, alongside the physical spherical-wave atom.Source: Appendix E · p. 82, §E.1 · p. 82

E.2. Jacobi-Anger identities and tail bounds.

• Jacobi-Anger tail bounds provide a deterministic truncation-error quantity that can be computed using certified interval bounds for Bessel functions.Source: §E.2 · pp. 82–83

E.4. Lifted Bessel-Vandermonde factorization.

• The lift groups retained harmonics as m = p + 2q and uses their aggregated coefficients in a Bessel-Vandermonde factorization.Source: §E.3 · p. 83, §E.4 · p. 85

E.3. Fresnel Bessel-Vandermonde expansion.

• The finite-harmonic approximation has an explicit atomwise error bound uniform over sensor index, range bin, and admissible angle, and the truncation error tends to zero as the orders increase.Source: §E.3 · p. 84

E.5. Physical spherical-wave atom and Fresnel remainder. √

• The physical spherical-wave path is expanded around the Fresnel path, with a uniform remainder bounded at cubic order in the aperture parameter.Source: §E.5 · p. 85

E.6. Physical atom approximated by the Fresnel Bessel-Vandermonde lift.

• The physical spherical-wave atom approximated by the Fresnel lift retains a paraxial mismatch as truncation orders increase; that residual need not vanish unless the paraxial error also vanishes.Source: Appendix E · p. 82, §E.6 · p. 86

E.7. Operator error bounds for semi-discrete measures.

• The deterministic approximation bounds separate harmonic-truncation error from paraxial spherical-wave error.The bounds concern the finite-harmonic lifted physical model.Source: §E.7 · pp. 86–87

Appendix F. Lifted atomic norm duality and SDP representation

• The lifted model is a continuous-angle, finite-range atomic norm problem, with atoms indexed by range bin and angle.The appendix derives its primal and dual problems, SDP representation, support-extraction condition, and perturbation bound to the Fresnel QPAC certificate.Source: Appendix F · p. 87

F.1. From the tensor Bessel lift to the harmonic lift.

• Centered and one-sided harmonic conventions differ by a unit-modulus factor, so conversion between them preserves total variation.The centered convention is used in the main text and the one-sided convention in the Toeplitz SDP.Source: §F.1 · p. 87

F.2. Lifted atomic norm.

• For the physical angular set, omitting localizing Toeplitz constraints gives the conservative full-circle formulation.The full-circle trigonometric moment norm has an exact Toeplitz SDP representation; exact sector-restricted constraints require additional localizing constraints.Source: §F.2 · p. 88

F.3. Primal and dual lifted problems.

• The lifted formulation includes noiseless and robust programs, with Fenchel duality used for the robust problem.Source: §F.3 · pp. 88–89

F.4. Fejér-Riesz dual SDP.

• For full-circle angles, Fejér-Riesz gives an exact SDP representation of the dual polynomial bound; applying that constraint to a restricted physical angle set is conservative.An exact restricted-angle representation requires localizing constraints.Source: §F.4 · p. 89

F.5. Proof of dual saturation.

• An active angular atom in a range block yields that block's discrete range estimate.Source: §F.5 · p. 90

F.6. Extracting (r, θ) from a lifted solution.

• The support of every norm-attaining primal decomposition lies in the dual contact set, which is approximated numerically by a near-contact set.Source: §F.6 · p. 90
• Angles can be extracted by refining dual-polynomial peaks or rooting the corresponding trigonometric polynomial; Toeplitz moment matrices offer a Vandermonde-decomposition route.The latter route applies when the moment matrix has rank below its harmonic dimension.Source: §F.6 · p. 91
• After support extraction, amplitudes are estimated by minimum-norm least squares, with exact noiseless fitting when the support dictionary has full column rank.Source: §F.6 · p. 91

F.7. Relation to the Fresnel TV certificate.

• As PJA and QJA increase without bound, the finite lifted Fresnel model converges uniformly to the semi-discrete model; at finite truncation it is a surrogate unless discarded harmonics vanish.The perturbation estimate connects the ideal QPAC certificate to its finite lifted polynomial, but does not assert equality between the finite SDP optimizer and the QPAC vector.Source: §F.7 · p. 91