Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra

• At a fixed scale, geometric samples reduce to a finite exponential sum with nodes q^αℓ, connecting the recovery problem to classical Prony methods.Operator-based and Banach-module extensions are also noted.Source: §1 · pp. 1–2
• Dilation covariance generates the Hankel moment structure before the sparse Mellin model is imposed, after which classical Prony factorization follows.The normalization removes a scalar multiplier to restore the additive index i + j.Source: §1 · p. 2
• Incommensurate logarithmic sampling steps remove global aliasing for complex exponents after cross-scale matching, unlike scales used only for conditioning or computational efficiency.Distinct effective amplitudes pair nodes directly; repeated amplitudes require a compatibility gap in the joint scale embedding.Source: §1 · p. 2
• The minimal Hankel pencil’s determinant gives regional node counts by winding, while Rouché bounds certify persistent counts under perturbations.Small contours localize isolated nodes; larger contours can certify unresolved cluster cardinality.Source: §1 · p. 2
• As q approaches 1, the Hankel determinant vanishes at an explicit order and the fixed-contour margin degenerates with a corresponding power of |log q|.The experiments examine this degeneration alongside multiscale de-aliasing, clustered spectra under noise, and distinct point- and contour-channel failures.Source: §1 · p. 2

• The factor q^κij in products of dilation operators prevents L(E_iE_jf) from depending only on i + j, obstructing Hankel indexing.The projective multiplication law describes the scalar multiplier in ordered products.Source: §2 · p. 3
• The unique entrywise normalization q^-κij cancels that multiplier and restores the additive index i + j.It is not an independent reweighting of the moments.Source: §2 · p. 4
• In the geometric realization, the abstract lifting becomes a Hankel matrix formed from geometric samples; point evaluation is used on the pointwise function class, not as a bounded functional on L2(R+, dx/x).The dilation representation itself is realized on L2(R+, dx/x).Source: §2 · pp. 4–5
• For distinct nodes and nonzero weights, the Prony polynomial is the minimal annihilating polynomial of the moment sequence.This identifies the minimal recurrence encoded by the finite-rank sequence.Source: §2 · p. 5
• The generalized eigenvalues of the matrix pencil, zeros of its characteristic polynomial, and zeros of the Prony polynomial are the same sampled nodes.Source: §2 · p. 6

• For complex exponents, one-scale nodes do not determine the exponent uniquely because the logarithm is multivalued and the sampling map is periodic.A fundamental strip selects a representative by prior restriction rather than resolving the ambiguity from data.Source: §3 · p. 6
• If at least one pair of logarithmic steps has an irrational ratio, the joint aliasing lattice is {0} and the multiscale map is injective on C.For commensurate steps, ambiguity is reduced but remains; injectivity then requires restricting the imaginary part to a fundamental strip.Source: §3 · p. 7
• Distinct effective weights pair the unordered node sets across scales directly; repeated weights require a compatibility gap to make the true matching uniquely minimal.Under the stated perturbed-cost condition, the true matching remains uniquely minimal.Source: §3 · p. 7
• A correctly paired noiseless node pair has zero compatibility cost whenever its exponent lies in the admissible set.Source: §3 · p. 7
• With two scales and 2r consecutive samples per grid, nonzero amplitudes, distinct exponents, and the stated matching conditions give unique recovery up to permutation.The minimal model order is r, and all exponents and amplitudes are uniquely determined.Source: §3 · p. 8
• For real exponents, one scale already gives injectivity; additional scales affect resolution, while for complex exponents they can affect both resolution and uniqueness.Source: §3 · p. 8

4.1 Hankel perturbation and scale separation

• A perturbation smaller than half the smallest nonzero singular value preserves a visible Hankel rank gap.This gives a sufficient perturbation bound for rank resolution.Source: §4.1 · p. 9
• Rank resolution depends on the weakest effective weight and the Vandermonde matrix's smallest singular value; clustered nodes worsen the latter at a power-law rate in relevant super-resolution regimes.Source: §4.1 · p. 9
• The scalar scale surrogate has maximum 1/(eB), but it does not optimize the full Vandermonde condition number.Sample length, dynamic range, the exponent interval, and noise can shift the best practical scale.Source: §4.1 · p. 9
• The square Hankel determinant degenerates as q approaches 1 under the stated assumptions of distinct exponents and nonzero weights.The determinant analysis identifies this as part of the singular regime for q → 1.Source: §4.1 · pp. 9–10

4.2 Argument-principle counting and Rouché stability

• The argument principle gives the number of pencil nodes inside a contour, counted with multiplicity.The count is obtained from the minimal Hankel pencil's characteristic.Source: §4.2 · p. 10
• Under the stated Rouché conditions, exact and perturbed characteristics have the same zero count inside the contour.For computation, use the rank-r projected pencil when m > r; the unreduced m × m exact pencil has identically zero determinant.Source: §4.2 · pp. 10–11
• When the Rouché condition holds on a small contour around a node, the observed characteristic has exactly one zero inside it.The result applies to the disk centered at the scale-dependent node described in the passage.Source: §4.2 · p. 13
• A contour can preserve a cluster's node count even when no set of single-node contours meets the perturbation criterion.The regional count can therefore remain informative after pointwise localization breaks down.Source: §4.2 · p. 13

5 Dual-channel recovery algorithm and numerical experiments

• The procedure combines scale-wise rank selection, projected-pencil point estimates, and contour counts; unresolved components remain regional clusters rather than receiving artificial pointwise exponents.Resolved components are matched across scales, and optional refinement uses variable projection or damped Gauss–Newton iteration.Source: §5 · pp. 13–14

5.1 Numerical experiment I: exact finite-rank recovery

• In the noiseless three-mode experiment, a sharp singular-value gap after the third mode identifies numerical Hankel rank r = 3.The projected pencil is then used for recovery.Source: §5.1 · p. 16, Figure 1 · p. 17

5.2 Numerical experiment II: degeneration as q →1 and scale selection

• The fitted determinant-degeneration slope is 11.995 versus the theoretical 12; the scale surrogate’s analytic maximizer is q* = e^-1/4 = 0.778801 versus grid maximizer 0.779419.The slope differs from the theoretical value by approximately 5.4 × 10^-3.Source: Figure 2 · p. 17, §5.2 · p. 18, Figure 3 · p. 18

5.3 Numerical experiment III: complex aliasing and two-scale de-aliasing

• On the first grid, aliased nodes differ by 2.48 × 10^-16, but on the second grid they differ by 0.8732; compatibility selects the true branch with error 1.11 × 10^-16.The correct branch is k = 0, while competing branches retain an O(1) residual.Source: §5.3 · p. 18, Figure 4 · p. 19

5.4 Numerical experiment IV: clustered spectra under noise

• At δ = 10^-6, joint refinement improves median exponent RMSE from 1.379 × 10^-1 to 1.040 × 10^-1; the better-separated scale also raises the rank-three singular value by more than two orders of magnitude.At δ = 10^-8, joint refinement is essentially indistinguishable from the well-separated single-scale reconstruction, and it is slightly less accurate at δ = 10^-7.Source: §5.4 · pp. 19–20, Table 1 · p. 20

5.5 Numerical experiment V: dual-channel contour recovery and certification

• With exact data, determinant phase accumulation gives count three at both scales; at δ = 10^-11, adaptive localization retains one count-three cell at q = 0.90 but resolves three singleton cells at q = 0.65.The point-channel estimates are overlaid for comparison and are not used in the winding computation.Source: §5.5 · pp. 20, 22, Figure 5 · p. 21, Figure 7 · p. 22
• At q = 0.65 and δ = 10^-7, winding counts give the correct cardinality in every trial while only 0.7% of point estimates meet the 10^-3 tolerance.At δ = 10^-8, pointwise success is 0.927, while winding counts are correct in all 150 trials and the numerical Rouché test passes throughout.Source: §5.5 · pp. 20, 22, Figure 6 · p. 21

• Covariance normalizes the projective multiplication law to restore additive Hankel indexing, with sparse-spectrum specialization yielding classical finite Prony factorization.Source: §6 · p. 22
• Across scales, incommensurate logarithmic steps remove complex aliasing after cross-scale matching, while well-separated scales improve clustered-spectrum resolution.Source: §6 · p. 22
• Minimal-pencil determinant winding counts nodes regionally without resolving them individually, and Rouché-type bounds give noise-dependent conditions under which counts persist.Source: §6 · p. 22
• 5.9637 is the fitted slope for the fixed-contour margin as q approaches 1, matching the predicted exponent r(r −1) = 6.Source: Figure 8 · p. 23
• Perturbation estimates are sufficient rather than sharp; quantitative clustered-spectrum resolution limits need finer multiscale Vandermonde analysis.Source: §6 · p. 23

• Here q is the Arik–Coon deformation parameter, not the Mellin sampling ratio; the appendix records a finite spectral problem and two-probe identification of q.Source: A A q-Fock realization · p. 23
• The analytic realization uses M_z F(z) = zF(z) and T_q F(z) = F(qz), with 0 < q < 1.Source: A A q-Fock realization · p. 23
• Each probe yields a finite-exponential structure with probe-dependent effective amplitudes; a sufficiently large associated Hankel matrix has rank equal to the number of visible distinct nodes.Source: A A q-Fock realization · p. 24
• One probe does not generally separate q from the coefficients, but common nodes across probes allow effective amplitudes to be matched by node.Source: A A q-Fock realization · p. 24
• Under the stated visibility conditions, every visible nonzero component gives the same q in the noiseless setting, and the ratio removes the superposition coefficient and coherent-state normalization.Source: A A q-Fock realization · pp. 24–25