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Numerical Stability of Extended-Domain Spectral Collocation: Floating-Point Regularization and an Equilibrium Law for the Frame Constants

Po-Yi Wu

arXiv:2610.13222v1math.NAcs.NA

TL;DR

Extended-domain collocation uses an exponentially ill-conditioned square system, despite observed geometric convergence. The paper explains this behavior through an operator-dependent interpolant, floating-point error bounds, and a potential-theoretic law for Poisson frame constants.

  • Problem

    How can an exponentially ill-conditioned square collocation system converge geometrically despite the impossibility theorem for equispaced data?

    Source: §10 · pp. 68–69

  • Method

    The paper identifies collocation with the solution operator applied to an interpolant in an operator-dependent test space and bounds coefficient error using two sampled-frame constants.

    Source: §2.4.3 · p. 13, §10 · p. 69

  • Results

    Under a condition on the perturbed matrix, a backward-stable solve attains the accuracy of frame approximation regularized at the backward-error level; for Poisson, the tail law starts with slope -2δ/(L+2δ).

    Source: §4 · p. 4, §10 · p. 69

  • Takeaways & Limitations

    Extension ratio, solution analyticity, and unit roundoff determine interpolatory accuracy; oversampled normalized truncated-SVD is preferred when seeking accuracy beyond its floor.

    Source: §10 · p. 70

    The explicit bounds are proved only for Poisson; their generalization to convection–diffusion remains open.

    Source: §2.4.6 · p. 21

Abstract

from arXiv · show

Extended-domain spectral collocation expands the solution of a boundary value problem in the Laplacian eigenbasis of an enclosing box and collocates at as many nodes as modes. The square system is exponentially ill-conditioned, and the impossibility theorem for equispaced data rules out stable geometric convergence; yet such solvers converge geometrically to a few orders of magnitude above the unit roundoff. We explain why in one dimension. The collocation solution is the solution operator applied to the interpolant of the data in an operator-dependent test space whose Lebesgue constant must grow exponentially for geometric convergence, so exact solutions can amplify data errors exponentially. In floating-point arithmetic the error of any coefficient vector is bounded by its nodal residual, its distance to a competitor and two frame constants, a leakage and a restricted Marcinkiewicz--Zygmund constant, bounded uniformly in the number of modes at a fixed positive threshold; for a backward-stable solve, under a condition on the perturbed matrix, the bound is uniform in the conditioning for suitably analytic data and, with computed constants, within a factor of about ten of the error floor of a direct solver. For the Poisson operator the rate of the two constants along the tail of the spectrum is given by a constrained equilibrium problem of logarithmic potential theory. For an interval of length $L$ extended by $\delta$ at each end, the problems for $\delta$ and $L^2/(4\delta)$ are dual, the rate is linear for $\delta = L/2$, and it starts with slope minus the fraction of the extended domain occupied by the extension, so for $\delta < L/2$ the leakage constant obeys no power law with the exponent of the end of the spectrum. In experiments, convection--diffusion operators behave like the Poisson operator at moderate P\'eclet numbers.

1. Introduction

Extended-domain collocation uses a square system that is exponentially ill-conditioned despite geometric convergence in reported examples. In one dimension, its exact solution can amplify data errors exponentially because geometric convergence in the operator-dependent test space requires an exponentially growing Lebesgue constant.

1. Introduction

  • The method collocates at as many equispaced nodes as modes, producing an exponentially ill-conditioned square system despite geometric convergence in reported examples.The reported examples include Poisson and convection–diffusion problems on a square and Poisson on a disk.Source: §1 · p. 2

1. The exact method is unstable.

  • The collocation solution applies the continuous solution operator to the data interpolant in the operator-dependent test space.That test space is the image of the trial space, which incorporates the boundary conditions, under the differential operator.Source: §1 · p. 2
  • Geometric convergence for all data analytic in a fixed neighbourhood requires an exponentially growing test-space Lebesgue constant, which can amplify data errors exponentially.For the Poisson operator, the test-space Lebesgue constant grows at the trial-frame rate without a convergence assumption.Source: §1 · pp. 2–3

2. Floating-point arithmetic regularizes.

Floating-point error is controlled by the nodal residual, distance to a competitor, and two sampled-frame constants; under a condition on the perturbed matrix, backward-stable solves can attain frame-regularized accuracy.

2. Floating-point arithmetic regularizes.

  • The error of any coefficient vector is bounded using its nodal residual, distance to a competitor, and the leakage and restricted Marcinkiewicz–Zygmund constants at threshold t.At a fixed positive threshold, these constants are bounded uniformly in the number of modes.Source: §2 · p. 3
  • A backward-stable solve attains frame-approximation accuracy regularized at its backward-error level if the perturbed matrix has no singular value far below that error.For sufficiently analytic data, the resulting error bound is polynomial in N and uniform in system conditioning.Source: §2 · p. 3
  • With exactly computed constants, the bound reproduces an LU solve’s error floor to within a factor of about ten.Source: §2 · p. 3

3. An equilibrium law determines the two constants.

For the Poisson operator, the leakage and restricted Marcinkiewicz–Zygmund constants along the spectral tail are described by a constrained logarithmic-equilibrium problem.

3. An equilibrium law determines the two constants.

  • At a fixed positive threshold, the leakage and restricted Marcinkiewicz–Zygmund constants remain bounded uniformly in N.Along the tail, the analysis considers thresholds t = e^(-Nτ), with 0 < τ < κσ.Source: §3 · p. 3
  • The tail rate is given by a constrained logarithmic-potential energy problem for measures bounded by the node density, with external field −τ on the physical interval.The equilibrium measure equals the node density on layers at the physical interval’s ends and leaves a gap beside them in the extension.Source: §3 · p. 3
  • The tail law characterizes extremal polynomials and optimal Lagrange interpolation sets, with matching lower and upper bounds.It is also the value of a two-level variant of Rakhmanov’s extremal problem for polynomials bounded on a discrete set [38].Source: §3 · pp. 3–4

4. The structure of the law decides the power law.

The extension-dependent equilibrium law determines when the leakage constant can follow a power law, while the paper also compares convection–diffusion and oversampled formulations and specifies the analysis’s scope.

4. The structure of the law decides the power law.

  • For δ = L/2, the tail law is linear and max{γ_t, tΓ_t} behaves like t^(1/2) at the exponential scale; its initial slope for general δ is −2δ/(L + 2δ).For δ < L/2, the leakage constant obeys no power law with the endpoint exponent; whether the law stays below that power-law rate throughout the tail for δ > L/2 remains open.Source: §4 · pp. 4–5
  • At moderate Péclet numbers, controlled experiments find that the convergence rate does not depend on the operator and the accuracy floor depends on it only weakly.The paper attributes the weak high-mode effect to a phase shift bounded by k(L + 2δ)/(jπ).Source: §4 · p. 4
  • Oversampling with truncated singular value decomposition gives a test-space Marcinkiewicz–Zygmund stability bound, and fourfold oversampling attains errors of a few tens of units of roundoff.The corresponding constant grows exponentially for the square system and far more slowly under fourfold oversampling.Source: §4 · p. 4
  • Unlike regularized frame approximation, the square collocation system is not covered by the cited theory, which requires truncation; oversampling supports accuracy at the truncation level.Prior work on polynomial frames establishes exponentially decaying error down to a prescribed tolerance under linear oversampling.Source: §4 · p. 4
  • The equilibrium framework extends earlier constrained-potential results by using an external field to determine sampled-frame stability constants along the spectrum’s tail.The related work traces constrained equilibrium problems to Rakhmanov and subsequent extensions.Source: §4 · p. 5
  • The two-constant analysis is one-dimensional, and the Poisson tail law gives exponential-scale rates rather than constant-threshold limits as N grows.Tensor-product domains are treated through an extension of the representation and ill-conditioning analysis; two-dimensional and disk cases are numerical.Source: §4 · p. 5

2. Setting and Main Results

The section develops the extended-domain collocation framework and relates geometric convergence in exact arithmetic to exponential data instability, while bounding floating-point error through nodal residuals and sampled-frame constants. It also establishes conditioning-independent bounds for analytic data and derives frame-constant tail laws, including explicit sharp bounds for the Poisson operator.

2.1. Basis and Discretization.

  • The extended-domain basis consists of Laplacian eigenfunctions on the larger interval, whose restrictions form a tight frame with asymptotic frame bound (L + 2δ)/L.This ratio measures frame redundancy and increases with the extension length δ.Source: §2.1 · pp. 6–7

2.2. Trial Space, Test Space, and the Constants of the Sampled Frame.

  • The sampled-frame constants separate poorly resolved coefficient directions, measured by leakage, from well-resolved directions, measured by the restricted Marcinkiewicz–Zygmund constant.For a fixed positive threshold, the cited a priori bounds give γ_t ≤ CΦ and Γ_t ≤ CΦ/t for sufficiently large N.Source: §2.2 · pp. 8–9, §2.4.3 · p. 14

2.3. The Chebyshev Variable and the Node Potential.

  • Under the cosine change of variables, the sine eigenbasis becomes weighted polynomial interpolation on the physical interval, with equispaced nodes inducing a limiting arcsine node measure.The extended interval maps to z ∈ (−1, 1), while the physical interval maps to z ∈ (−c0, c0).Source: §2.3 · pp. 9–10

2.4.1. Exact Arithmetic.

  • Geometric convergence forces the test-space Lebesgue constant to grow exponentially, so the exact solution can exponentially amplify data errors.For the Poisson operator, the test-space Lebesgue constant grows at the trial-frame rate without a convergence hypothesis, and unisolvence holds for all sufficiently large N.Source: §2.4.1 · p. 11

2.4.2. Rates of the Trial Frame.

  • The frame exponents obey a(ω) + a(π−ω) = 1, and the Poisson bulk collocation condition number grows at least as C N^-3/2 e^(αN) for every α < κσ(ω).At δ = L/2, a(π/2) = 1/2; the duality exchanges the physical-domain and extension arcs.Source: §2.4.2 · p. 12

2.4.3. Floating-Point Arithmetic.

  • A two-constant bound controls coefficient-vector error through the nodal residual, distance to a comparator, leakage, and the restricted Marcinkiewicz–Zygmund constant.For backward-stable solves, the bound applies when the perturbed matrix satisfies the stated lower bound on its minimum singular value; the direct solver then attains the accuracy of an η-regularized approximation.Source: §2.4.3 · p. 13
  • For sufficiently large N, the two-constant framework bounds the leakage uniformly and the restricted constant by CΦ/t, providing size controls for the floating-point error estimate.The bounds hold for N beyond a threshold depending on δ, L, and k.Source: §2.4.3 · p. 14
  • At fixed positive thresholds, the leakage and restricted Marcinkiewicz–Zygmund constants are uniform in mode count, so a bounded-norm competitor matching the data to backward-error accuracy yields an error bound uniform in conditioning.The Lebesgue-type bounds instead grow with the test-space Lebesgue constant.Source: §2.4.3 · p. 14

2.4.4. Analytic Data.

  • For sufficiently analytic data, backward-stable solves converge geometrically until an η^a′ error floor takes over, with a bound polynomial in N and uniform in system conditioning.The exponent is a′ = log ρ/log ρ*; it equals 1 when the data are analytic on the largest region Ωρ*.Source: §2.4.4 · pp. 14–15

2.4.5. The Tail of the Spectrum.

  • At thresholds t_N = e^-Nτ, the paired frame-constant rate is governed by a constrained logarithmic-energy equilibrium problem whose unique minimizer determines the tail law.The tail law determines max{γ_t, tΓ_t} up to a factor e^o(N).Source: §2.4.5 · pp. 16–17
  • Complementary extensions δ and L^2/(4δ) have dual tail laws; for the self-dual extension δ = L/2, r(τ) = −τ/2 and the constants scale as t_N^(1/2+o(1)).The duality applies across the tail, while the self-dual law is linear.Source: §2.4.5 · pp. 17–18
  • The tail begins with slope −2δ/(L+2δ), the negative fraction of the extended domain occupied by the extension; for δ < L/2, leakage has no sharp power law with the endpoint exponent.For δ > L/2, the tail initially lies below the chord joining the endpoint rates.Source: §2.4.5 · pp. 18–19

2.4.6. Explicit Bounds Along the Tail.

  • For the Poisson operator, polynomial-zero and Lagrange-interpolation constructions give explicit sharp lower and upper bounds along the tail; extending these bounds to convection–diffusion remains open.The lower and upper bounds are based on zero-distribution and interpolation configurations, respectively.Source: §2.4.6 · pp. 19–21

3. Exact Structure of the Interpolatory Method: Proofs of Theorem 2.8 and Proposition 2.10

The collocation solution is the boundary-value solution operator applied to data interpolated in an operator-dependent test space. This representation explains both exponential amplification of nodal-data errors and how the operator affects the method’s behavior.

3. Exact Structure of the Interpolatory Method: Proofs of Theorem 2.8 and Proposition 2.10

  • The exact-structure proof works for continuous variable convection coefficients, while the nonsingularity assumption is necessary for the interpolatory representation.For fixed extension length, the Poisson assumption holds for all sufficiently large N; for fixed N, its possible failures are discrete extension parameters unless an integral vanishes identically.Source: §3 · pp. 22–24

3.1. The Representation Theorem

  • u_N = L^-1 P_N f, where P_N interpolates the data in Ψ_N = L(Φ_N), the image of the boundary-constrained trial space.The representation also applies to arbitrary interior nodal data; nonzero boundary values are handled by subtracting a trial-space function with the prescribed boundary values.Source: §3.1 · p. 24

3.2. What the Operator Contributes

  • For N much larger than k times the extended-domain length divided by π, convection changes high test-space modes only by small phase shifts, and experiments find nearly the Poisson Lebesgue-constant rate for k ≤ 50.At N ≲ k times the extended-domain length divided by π, convection reshapes the whole test space; experiments up to k = 10^5 report unchanged convergence rate but a higher unnormalized-system floor.Source: §3.2 · p. 25

3.3. Exact Arithmetic: Impossibility, Amplification of Data Errors, and the Discrete Green’s Function

  • Geometric interpolation convergence for all data analytic in a fixed neighborhood requires an exponentially growing test-space Lebesgue constant, which can amplify data errors enough to cause divergence beyond a precision-dependent threshold.For the Poisson operator, the amplification factor Γ_0 grows exponentially; rounded data eventually dominate the error bound, whereas exact data avoid this effect.Source: §3.3 · p. 26
  • The discrete Green’s-function rows follow the Lagrange-function mass profile, not the continuous kernel’s hat shape or convection–diffusion upstream decay.The masses are exponentially large, and the apparent row decay away from the diagonal reflects their decay toward boundary nodes.Source: §3.3 · p. 26

4. Potential Theory in the Chebyshev Variable: Proof of Theorem 2.12

Logarithmic potential theory gives exponential rates for the trial frame’s Lebesgue constant and smallest singular value. Those rates follow from the node potential in the Chebyshev variable and determine the method’s baseline ill-conditioning.

4. Potential Theory in the Chebyshev Variable: Proof of Theorem 2.12

  • The node potential is even in z and decreases with |z|, attaining its maximum at z = 0 and its minimum on the physical interval at ±c_0.The minimum on the full interval [−1, 1] is at ±1; the Lagrange-function norm estimates use this potential to obtain the frame rates.Source: §4 · pp. 27–28
  • Unlike full-interval equispaced polynomial interpolation, the frame’s node interval is replaced by a larger interval, producing two exponential rates rather than one.Full-interval equispaced interpolation has Lebesgue-constant growth rate 2^N.Source: §4 · p. 29

4.1. Exponential Ill-Conditioning

  • The frame’s smallest singular value decays exponentially at rate κ_σ, while its Lebesgue constant grows exponentially at rate κ_Λ, determining the baseline ill-conditioning of the Poisson collocation matrix.The frame consists of odd Fourier modes restricted from the doubled extended domain to a subinterval; restricted-system singular values decay beyond a transition index proportional to N L / L̃.Source: §4.1 · p. 30

5. Floating-Point Arithmetic: Proof of Theorem 2.16

The floating-point analysis bounds solution error using the residual, distance to a competitor, and two frame constants. It interprets a backward-stable computed solution as exact collocation on perturbed data, while analytic competitors support accuracy bounds.

5.1. Backward-Stable Solvers

  • A backward-stable solve of the recombined system is exactly interpolatory collocation for perturbed nodal data; the recombined system enforces boundary conditions by construction.For normalized columns, Householder QR has a backward-error bound independent of the conditioning of the normalized matrix.Source: §5.1 · pp. 31–32

5.2. The Two-Constant Bound

  • The error of any coefficient vector is bounded using its nodal residual, distance to a competitor, and leakage and restricted Marcinkiewicz–Zygmund constants at a chosen threshold.At fixed positive threshold, both constants are bounded uniformly in the number of modes; the backward-stable-solver bound additionally assumes a lower bound on the perturbed matrix’s smallest singular value.Source: §5.2 · pp. 32–33

5.3. Competitors of Moderate Coefficient Norm: Proof of Theorem 2.20

  • An analytic solution can be approximated by a boundary-corrected trial-space competitor whose residual on the physical interval decreases geometrically, up to polynomial factors.The construction approximates the solution and its first two derivatives in the Chebyshev variable, then corrects boundary values using the two lowest modes.Source: §5.3 · pp. 33–35
  • The competitor bound and two-constant estimate combine to control computed-solution error for backward error η, with the stated bounds depending on η and the competitor’s approximation quality.The proof uses analytic continuation and Chebyshev expansions; the resulting bounds include polynomial factors in the approximation degree.Source: §5.3 · pp. 34–35

6. Proofs for the Tail of the Spectrum

This section proves tail-spectrum bounds for the frame constants, connecting their rates to an equilibrium problem and an extremal zero-distribution problem, and establishing a duality between the rate functions. It derives a linear tail rate up to an O(τ^1.5) error, with slope set by the extension fraction, and uses these results to address power-law bounds and the equilibrium measure near the interval endpoint.

6.1. The Equilibrium Problem: Proof of Proposition 2.23

  • For τ < κ_σ, the unique symmetric equilibrium has positive mass on the extension and a layer structure; for τ ≥ κ_σ, it equals the node measure.The layer consists of symmetric endpoint intervals with a unique boundary parameter θ_τ.Source: §6.1 · pp. 37–39

6.2. Configurations and Lattice Products

  • Away from layer nodes, lattice points, and π/2, (N−1)^−1 log|P_N(cos θ)| ≥ E(θ) − C log N/N.The bound assumes θ stays at least Δ/4 from each layer node, d_k/4 from each lattice point, and 1/N from π/2.Source: §6.2 · p. 40

6.3. From Polynomials to Coefficient Vectors

  • Polynomial-potential estimates yield exponential coefficient-vector bounds with exponent gap κ_σ − κ_Λ = U_ω(c_0) − U_ω(1).The proof also bounds the associated Lagrange functions and derives the corresponding threshold behavior.Source: §6.3 · pp. 41–42

6.4. Proof of Theorem 2.34 and of Corollaries 6.6 and 6.7

  • The proof of Theorem 2.34 starts from a symmetric zero polynomial, corrects its function into Ψ_N, and forms a coefficient vector whose values match at interior nodes.It uses g_0 = sin θ P_N(cos θ) and a correction by χ_N^±.Source: §6.4 · p. 43
  • For configurations with ℓ(C) > −(κ_σ−κ_Λ), the nodal-level bound forces at least one of γ_t and tΓ_t to grow exponentially when ℓ(C) is positive.At t = e^-Nτ, the proof gives e^-Nℓ(γ_t + tΓ_t) ≥ e^-C log N.Source: §6.4 · p. 44
  • If ℓ(C) > −(κ_σ−κ_Λ) and ℓ(C)+a(ω)τ(C) > 0, the sharp power law γ_t ≤ Ct^a(ω) fails.The failure holds for all sufficiently large N under these configuration conditions.Source: §6.4 · p. 44

6.5. The Upper Bound: Proof of Theorem 2.36

  • When A1(S)/(1−a) ≤ −A2(S)/a, the certified bounds are γ_t ≤ 2NCt^a and Γ_t ≤ 2NCt^(a−1) at t = e^-Nτ.These bounds hold for every N ≥ N0 and the τ values specified by Theorem 2.36.Source: §6.5 · p. 46

6.6. The Tail Law: Proofs of Theorem 2.24, Proposition 2.25 and Corollaries 2.26 and 2.37

  • The exponential rates of max{γ_t, tΓ_t} at t = e^-Nτ equal r_ω(τ), the value of an extremal problem over zero distributions.The extremal problem constrains the zero distribution and the polynomial’s size on the extended interval and a free set.Source: §6.6 · pp. 46, 49
  • Configurations and interpolation configurations can approximate r_ω(τ) from below and above, respectively, to arbitrary accuracy.For every ε > 0, there are configurations with τ(C) ≥ τ−ε and ℓ(C) ≥ r_ω(τ)−ε, and interpolation configurations with Φ(S,τ) ≤ r_ω(τ)+ε.Source: §6.6 · p. 48
  • The lower bound on Γ_0 has exponential rate at least κ_Λ as N tends to infinity.This follows by applying the tail-rate result and letting τ approach κ_σ.Source: §6.6 · p. 49

6.7. Duality: Proof of Theorem 2.27 and Corollary 2.28

  • The duality relates the normalized rate functions, and for ω = π/2 it gives r_π/2(τ) = −τ/2.The relation applies to the dual parameters ω and ω′, with normalized levels scaled by κ_σ.Source: §6.7 · p. 51

6.8. The Start of the Tail: Proof of Theorem 2.30 and Corollaries 2.31 and 2.32

  • For small τ, r_ω(τ) lies between −sτ−Cτ^1.5 and −sτ+Cτ^1.5, establishing the tail’s linear rate.Here s=ω/π=L/(L+2δ).Source: §6.8 · p. 53
  • When δ<L/2, the leakage constant cannot obey a power-law bound with the exponent a at the end of the spectrum; when δ>L/2, r_ω(τ)<−aτ for sufficiently small τ.The first conclusion follows by contradiction from the assumed bound γ_t≤C*t^a; the second uses a<s.Source: §6.8 · p. 53
  • For every τ in the admissible range, −τ<r_ω(τ)<0, and the equilibrium measure vanishes just left of θ0 and equals ω^-1 dθ just right of it.Consequently, θ_τ>θ0.Source: §6.8 · p. 53

7. Regularized Oversampled Collocation

Theorem 7.2 bounds the PDE error of regularized oversampled collocation through regularized best approximation in the test space and the constant C_M. The bound's usefulness depends on controlling this constant, including uniformly in the number of modes.

7. Regularized Oversampled Collocation

  • When C_M≤C*, the oversampled error bound holds uniformly in N; C_M is a test-space Marcinkiewicz–Zygmund constant computable from a generalized eigenvalue.The node matrix must have full column rank.Source: §7 · pp. 53–54
  • The PDE error is bounded by the data's ε-regularized best-approximation error in the test space, transferred through L^-1 and C_M.The theorem's content depends entirely on C_M.Source: §7 · pp. 53–54
  • With M=n and 0<ε<σ_n, regularized oversampled collocation equals the interpolatory solution and its bound uses C_n.Letting ε tend to zero for fixed competitor removes the regularization term.Source: §7 · p. 54

8. Higher Dimensions and Curved Boundaries

In higher dimensions, the frame instability compounds exponentially with dimension. Curved-boundary node relocation changes the sampling set but is not covered by the exact-representation theorem.

8. Higher Dimensions and Curved Boundaries

  • The multidimensional trial space uses a tensor-product grid, and its synthesis matrix is the Kronecker product of the one-dimensional synthesis matrices.For equal extension and interval lengths, the bulk Poisson matrix also factors across dimensions.Source: §8 · p. 55
  • For every α < κσ(ω), the condition number is at least C N^-3/2 e^(αdN) for all N ≥ 2, showing exponential compounding with dimension.The bound holds when the extension and interval lengths are equal across dimensions.Source: §8 · p. 55
  • The interpolatory solution is uniquely characterized by L uN = P_N,d f and zero boundary values when interior-node interpolation in Ψ_N,d is unisolvent.The theorem assumes a continuous drift coefficient and a solution in C²(R0) ∩ C(closure R0).Source: §8 · p. 55
  • The multidimensional solution error is bounded by the forcing interpolation error, with the Lebesgue constant controlling its approximation contribution.The proof gives an infinity-norm bound of (e^(αL1) - 1) times the forcing interpolation error, which is at most (1 + Λ(Ψ_N,d)) dist∞(f, Ψ_N,d).Source: §8 · p. 56
  • For constant drift, the test spaces of the Poisson and convection–diffusion operators are heuristically expected to have nearly the same high modes.This heuristic follows from applying Lemma 3.3 direction by direction for constant k.Source: §8 · p. 56
  • Node relocation for curved boundaries moves nearby grid nodes onto the boundary, but the resulting exterior problem has no boundary condition and is not covered by Theorem 8.2.The forcing must be defined throughout R0, requiring a smooth extension for general data.Source: §8 · p. 56

9. Numerical Experiments

The experiments examine floating-point behavior, solver variants, frame bounds, and potential-theoretic rates across one- and two-dimensional settings. They find shared convergence patterns across operators in the tested regimes, while oversampling and the two-constant bound clarify accuracy floors and spectral-tail behavior.

9.1. Controlled One-Dimensional Comparison

  • Rounded-data exact solves diverge beyond N*≈40, while exactly evaluated data converge below 10^-16 at N=64; geometric-phase rates coincide across operators for each δ.The double-precision LU solve also diverges after the same threshold, whereas coefficients with exactly evaluated data remain size 1.4 at N=64.Source: §9.1 · p. 57

9.2. The Exact Discrete Green’s Function

  • The discrete Green’s-function row exceeds the continuous kernel by up to seven orders for Poisson and nine for k=20, tracking Lagrange-function masses that reach 2.4 × 10^8 centrally.The Poisson and convection–diffusion rows nearly coincide even where their continuous kernels differ by up to four orders of magnitude.Source: §9.2 · pp. 58–59, Figure 3 · p. 59

9.3. Solvers, Oversampling, Marcinkiewicz–Zygmund Constants, and Large Péclet Numbers

  • Fourfold oversampling reaches a 2.5 × 10^-15–7 × 10^-15 floor, over three orders below the interpolatory floor at δ=0.5; normalized TSVD matches unregularized solves.Across the tested range, fourfold oversampling keeps the Marcinkiewicz–Zygmund constant moderate, but fixed-factor oversampling is not expected to bound it generally; normalized oversampling also maintains a 3 × 10^-15–7 × 10^-15 floor for k up to 10^5.Source: §9.3 · p. 60

9.4. Evaluation of the Two-Constant Bound

  • At t=10^-15 and N≥48, neither frame constant grows with N, and the minimized two-constant bound tracks the LU error within factors of 4.4–8.5 at the floor.The bound also reproduces the flat error floor; convection–diffusion tail exponents differ from Poisson by at most 0.004 in the tested cases.Source: §9.4 · pp. 60–62

9.5. Potential-Theoretic Rates

  • At τ/(κσ)=0.02 and 0.05, computed start-of-tail ratios rω(τ)/τ fall in [−0.342,−0.329], [−0.669,−0.658], and [−0.802,−0.786] for δ=L/4, L, and 2L, respectively.Fitted spectral-tail exponents lie within 0.02 of a(ω) wherever the tail contains more than thirty directions.Source: §9.5 · pp. 63, 66

9.6. Certified Configurations on Both Sides of the Critical Extension

  • Certified configurations show the sharp power law fails at δ=L/4, while for δ=2L the constants satisfy the stated tail bounds over τ∈[0.080, 0.960]κσ.Whether the sharp power law holds for δ>L/2 with an N-independent constant at all levels remains undecided.Source: §9.6 · pp. 66, 68

9.7. Controlled Two-Dimensional Comparison on the Box

  • On the two-dimensional box, the tested operators share geometric-phase histories; for δ=1 and 2 their error floors lie within a factor of about 25.Saturation begins at about the same N as in the one-dimensional comparison.Source: §9.7 · p. 68, Figure 8 · p. 68

9.8. A Disk with Relocated Boundary Nodes

  • On the disk with relocated boundary nodes, convergence is geometric on average but nonmonotone, and the two operators’ smallest errors are within a factor of five for δ=1 and 2.The computations report that relocation preserves the frame’s floating-point behavior.Source: §9.8 · p. 68

10. Conclusion

The conclusion links interpolatory collocation to test-space interpolation, explaining exponential data amplification alongside floating-point stability. It summarizes a Poisson tail law and practical implications while identifying limits on the a priori theory.

10. Conclusion

  • The collocation solution applies the continuous solution operator to test-space interpolation, whose Lebesgue constant must grow exponentially for geometric convergence and can amplify data errors.For Poisson, that constant grows at the explicit rate of the trial frame.Source: §10 · p. 69
  • A backward-stable solve attains regularized-frame approximation accuracy if the perturbed matrix has no singular value far below the backward error.The error is controlled by two sampled-frame constants.Source: §10 · p. 69
  • For Poisson, a constrained logarithmic-potential equilibrium problem gives the two constants’ exponential-scale tail rates, with dual extensions δ and L^2/(4δ).The rate is linear for δ = L/2 and starts with slope equal to minus the extended-domain fraction.Source: §10 · p. 69
  • For δ < L/2, the leakage constant does not obey the power law with the exponent at the spectrum’s end.The tail law gives constants at exponentially decreasing thresholds, not at a fixed threshold that sets the accuracy floor.Source: §10 · p. 69
  • Extension ratio, solution analyticity, and unit roundoff determine the interpolatory method’s accuracy; oversampled normalized truncated SVD is preferred beyond its accuracy floor.The operator enters at large Péclet numbers only through the matrix norm.Source: §10 · p. 70
  • The tail law is proved only for Poisson; its analysis is one-dimensional, and convection–diffusion and tensor-product extensions remain incompletely analyzed.The computed convection–diffusion tail agrees, but its proof is open.Source: §10 · p. 70

A. Standard Facts

The appendix gathers standard tools for one-dimensional and box Dirichlet problems, Green functions, Laplacian eigenfunctions, and regularized least squares.

A. Standard Facts

  • The one-dimensional Dirichlet problem has a maximum principle, yielding uniqueness for zero boundary values, and its Green function represents the solution and supports inverse-norm bounds.The appendix also gives explicit Green functions for the Poisson and constant-convection–diffusion model operators.Source: A. Standard Facts · pp. 71–72
  • On a bounded set contained in a slab, the box Dirichlet lemma gives a maximum principle and bounds the solution’s supremum norm by the forcing norm.It also establishes uniqueness for the stated operator and boundary conditions.Source: A. Standard Facts · p. 72
  • On a box, Dirichlet Laplacian eigenfunctions form an orthonormal basis, and Green’s identity relates coefficients of −Δv to those of v.The resulting coefficient relation uses the eigenvalues of −Δ.Source: A. Standard Facts · p. 73
  • The appendix records elementary periodicity, symmetry, duplication, and endpoint expansions for the Clausen function, plus a logarithmic cosine integral identity.These properties are included with short proofs or references to earlier equations.Source: A. Standard Facts · p. 73
  • The regularized least-squares residual splits into projected and complementary parts, with bounds controlled by the competitor norm and the inverse regularization threshold.The complementary contribution from the retained singular vectors is bounded by ε^-1 times the competitor residual.Source: A. Standard Facts · p. 74

B. Norms of the Lagrange Functions of the Trial Frame: Proof of Proposition 4.3

The proof estimates discrete logarithmic potentials on the sampled nodes, then uses those estimates to bound Lagrange functions and their norms on the extended and physical intervals.

B. Norms of the Lagrange Functions of the Trial Frame: Proof of Proposition 4.3

  • A logarithmic-singularity Riemann-sum estimate controls discrete potentials uniformly, including sums with a node removed and points near a singularity.The proof separates cells near the singularity, away from it, and at the interval ends.Source: B. Norms of the Lagrange Functions of… · pp. 74–75
  • The Lagrange-polynomial bounds compare their logarithmic magnitudes with differences of the continuous potential, with O(log N) errors in the exponent.A separate lower bound applies when the evaluation point stays at least Δ/4 from every node.Source: B. Norms of the Lagrange Functions of… · pp. 76–77
  • The potential’s modulus of continuity controls its variation over short angular intervals, including intervals containing its singular point.The proof uses symmetry and integrates a logarithmic derivative bound.Source: B. Norms of the Lagrange Functions of… · p. 77
  • Upper and lower integral estimates for Lagrange functions on the extended and physical intervals convert potential differences into exponential-scale norm bounds.The change of variables contributes only O(log N/N) to the rates.Source: B. Norms of the Lagrange Functions of… · pp. 77–78

C. Column Norms of the Collocation Matrix: Proof of Proposition 2.18

The proof establishes uniform nodal lower bounds for shifted sines, uses them to bound collocation-matrix column norms, and applies these estimates to the sampled-frame constants.

C. Column Norms of the Collocation Matrix: Proof of Proposition 2.18

  • Shifted sines have nodal squared norms bounded below proportionally to N, uniformly over the phase and frequencies in the stated ranges.The proof treats large frequencies, intermediate frequencies, and the finitely many smaller frequencies separately.Source: C. Column Norms of the Collocation… · pp. 78–79
  • The collocation-matrix column norms are bounded above and below by positive constants depending only on the stated domain and operator parameters.The proof combines eigenvalue bounds with the shifted-sine nodal estimates.Source: C. Column Norms of the Collocation… · pp. 78–80
  • The trial-basis Gram matrix is I + aa^T + bb^T, and the column-norm bounds support uniform-in-N bounds for the frame constants.The proof uses the bounded basis coefficients and the diagonal scaling derived from column norms.Source: C. Column Norms of the Collocation… · p. 80

D. Technical Lemmas for Theorems 2.20–2.36 and the Certified Configurations

The appendix supplies technical estimates used by Sections 6 and 9, including bounds for envelopes, coefficient norms, test functions, and interpolation configurations. These estimates support the certified configuration evaluations.

Arc estimates

  • A Riemann-sum estimate on an interior arc controls the logarithmic kernel when node values approximate its integral.Its assumptions include an arc contained in [ξ, π − ξ] and sufficiently small lattice spacing.Source: D. Technical Lemmas for Theorems… · pp. 80–81

Envelope estimates

  • The envelope of a configuration has a modulus-of-continuity bound derived from the logarithmic kernel, allowing its values to be controlled between evaluation points.For separations s ≤ 1/2, the stated bound is 4s(1 + log(2π/s)).Source: D. Technical Lemmas for Theorems… · p. 81

Norm and test-function estimates

  • The appendix gives converse norm estimates relating test functions and coefficient vectors, and a lower bound for a test function when its second derivative has constant sign.For the latter bound, the second derivative must have magnitude at least g* on an interval of length h.Source: D. Technical Lemmas for Theorems… · pp. 82–83

Interpolation configurations

  • Lagrange polynomials for interpolation configurations are bounded using their envelope, and the constructed point sets have between N and N + K0 distinct points.The bound is log |ℓ_s(cos θ)| ≤ N[E(θ) − E(s)] + C log N.Source: D. Technical Lemmas for Theorems… · pp. 83–85

D.1. The Certified Configuration

The appendix certifies a configuration for δ = L/4 and describes procedures for certifying interpolation configurations. It also proves the boundary-layer structure underlying the start of the tail law.

D.1. The Certified Configuration

  • For δ = L/4, the certified configuration has ℓ + aτ ≥ 0.0084, with −ℓ/τ = 0.378 against a = 0.4.Its extension arcs end before θ0, and the total mass is 0.995.Source: §D.1 · p. 86, Table 2 · p. 86
  • The certified configuration was found by a linear program over piecewise-constant densities, with constraints sampled on the arcs and a prescribed mass bound.The arcs were refined near θ0 and θs, and θs was scanned.Source: §D.1 · p. 86

D.2. The Certified Interpolation Configurations

  • The interpolation certificates bound the quantities A1(S) and A2(S), yielding certified intervals for levels τ that lie within the stated range.The padded bounds determine intervals contained in (6.11).Source: §D.2 · pp. 87–88
  • Certified interpolation configurations are built from piecewise-constant densities and their certified intervals are combined into subcovers of the ranges in Corollary 9.2(ii).Table 3 covers δ = L with 52 of 76 certificates, while Table 4 covers δ = 2L with 25 of 40.Source: §D.2 · p. 88, Table 3 · p. 88, Table 4 · p. 89

E. The Boundary Layer at the Start of the Tail

  • Near the physical-interval endpoint, the boundary-layer density vanishes on a gap and equals the node density on a layer, while the envelope differs by τ between extension and physical regions.The density and envelope are related by a Hilbert transform in the stated limiting boundary-layer description.Source: E. The Boundary Layer at the Start of… · pp. 90–91
  • The boundary-layer envelope is continuous and nonincreasing across the interval, with endpoint values 0 at −g and −1 at l.Its derivative is Hölder continuous with exponent 1/2 on compact subsets away from zero.Source: E. The Boundary Layer at the Start of… · p. 91
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