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Error estimates for vector field interpolation based on generalized matrix-valued kernels

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Pith's one-line read This paper builds divergence-free and curl-free matrix-valued kernels from a single rough radial profile $\beta=D^k\psi_m$, identifies their native spaces with vector-valued Sobolev spaces, and proves sharp direct and inverse error…

desk verdict Useful, well-written kernel construction paper with a real proof gap in the main native-space theorem; the gap is fixable and the numerics stand. read the letter →

arxiv 2608.04313 v1 pith:6C6FG373 submitted 2026-08-05 math.NA cs.NA

Zhengjie Sun , Lishuo Dong , Leevan Ling This is my paper · ORCID
classification math.NAcs.NA MSC 41A2541A3565D0565D12
keywords matrix-valuedkernelsdivergence-freeinterpolationcurl-freenativespaceSoboleverrorestimatesBernsteininequalitydimension-walkingoperatorsscattereddataapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that an operator-based recipe—take the isotropic matrix kernel $K(x,y)=\alpha(r)I+\beta(r)(x-y)(x-y)^\top$ and set $\beta=D^k\psi_m$—produces divergence-free and curl-free interpolants whose native spaces are exactly vector-valued Sobolev spaces. The main results are a norm equivalence $\mathcal{N}_K(\mathbb{R}^d)\cong\widetilde{H}^{m+1-k}(\mathbb{R}^d)$ (Theorem 4.1) and the sharp error bound $\|f-I_X f\|_{W^\mu_q(\Omega)}\le C h_{X,\Omega}^{m-\mu-d(1/2-1/q)_+}\|f\|_{H^m(\Omega)}$ for divergence-free targets with Sobolev regularity $m$ (Theorems 4.6 and 4.7). The practical point is that enforcing conservation laws such as incompressibility during scattered-data approximation no longer forces a highly smooth scalar potential: the $k=0$ construction works with merely continuous integrable seeds, and Bernstein-type inequalities plus the inverse theorem close the theory. Numerical experiments with Matérn and Wendland kernels confirm the predicted rates $O(h^{4.5})$, $O(h^{3.5})$ and $O(h^{2.5})$ for $k=0,1,2$.

What carries the argument

The load-bearing mechanism is a pair of operators on radial functions: the integral operator $(I\phi)(r)=\int_r^\infty t\phi(t)\,dt$ and the differential operator $(D\phi)(r)=-\phi'(r)/r$, together with their dimension-walking relations between radial Fourier transforms in dimensions $d$, $d-2$ and $d+2$. These operators convert the ordinary differential equation that enforces the divergence-free condition into the closed-form coefficient rule $\alpha=(d-1)I\beta-r^2\beta$, and each application of $D$ shifts the Fourier decay exponent by one. The resulting matrix kernel has symbol $\widehat K(\xi)=\|\xi\|^2F_{d+4-2k}\psi_m(\|\xi\|)I$, whose spectral decay is exactly what the native-space-to-Sobolev norm equivalence (Theorem 4.1) exploits; the Bernstein inequality for band-limited trial functions then converts this identification into inverse estimates.

What would settle it

Compute $F_{d+4-2k}\psi_m(\omega)(1+\omega^2)^{m+2-k}$ on $\omega>0$ for a positive-definite radial kernel $\psi_m$ that satisfies (4.1) but is chosen outside the Matérn and Wendland families. If this ratio is not bounded above and below by positive constants, the dimension-walking step in Theorem 4.1 fails, and the claimed native-space identification together with the Sobolev rates of Section 4 collapses.

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Extended reading notes

Core claim

The central discovery is that the divergence-free constraint on $K(x,y)=\alpha(r)I+\beta(r)(x-y)(x-y)^\top$ reduces the two coefficient functions to one: $\alpha=(d-1)I\beta-r^2\beta$ with $(I\beta)(r)=\int_r^\infty t\beta(t)\,dt$, and the curl-free constraint similarly reduces to $\alpha=-I\beta$. Taking $\beta=D^k\psi_m$ with $D\phi=-\phi'/r$ and using the dimension-walking identities $F_d(\phi)=F_{d-2}(I\phi)$ and $F_d(\phi)=F_{d+2}(D\phi)$ gives the Fourier symbol $\widehat K(\xi)=\|\xi\|^2F_{d+4-2k}\psi_m(\|\xi\|)I$. When $F_d\psi_m(\omega)\asymp(1+\omega^2)^{-m}$, the paper identifies the native space with $\widetilde{H}^{m+1-k}(\mathbb{R}^d)$ under equivalent norms and splits it into the divergence-free and curl-free subspaces. On a bounded Lipschitz domain with quasi-uniform nodes it then proves $\|f-I_X f\|_{W^\mu_q(\Omega)}\le C h_{X,\Omega}^{m-\mu-d(1/2-1/q)_+}\|f\|_{H^m(\Omega)}$ for divergence-free $f\in H^m(\Omega)$; for targets with fractional regularity $\tau$ outside the native space the rate becomes $h^{\tau-\mu-d(1/2-1/q)_+}\rho_{X}^{m-\tau}$. The Bernstein inequality $\|u\|_{H^\mu(\Omega)}\le Cq_X^{-\mu}\|u\|_{L^2(\Omega)}$ on the trial space yields the complete inverse theorem, and the smallest eigenvalue of the interpolation matrix is bounded below by $c_d q_X^{2m-d-2k+2}$.

Load-bearing premise

The whole Section 4 chain rests on the assumption that when a scalar kernel $\psi_m$ is transferred from dimension $d$ to dimension $d+4-2k$, its Fourier-transform decay exponent increases by exactly $2-k$; Matérn and Wendland kernels have this property, but the stated decay condition (4.1) alone does not imply it for a general kernel.

Editorial extensions

If this is right

  • For a fixed scalar seed $\psi_m$ satisfying (4.1), the choice $k=0$ yields a divergence-free kernel whose native space is $\widetilde{H}^{m+1}(\mathbb{R}^d)$, improving the rate of the classical $k=2$ potential construction by $h^2$; the experiments measure $O(h^{4.5})$ versus $O(h^{2.5})$ for Matérn $\phi_{7/2}$.
  • The $k=0$ recipe requires no differentiation of the scalar seed, so continuous, integrable, or compactly supported radial functions become admissible generators of divergence-free interpolation.
  • For divergence-free targets of Sobolev order $m$, the interpolation error on a quasi-uniform set is $O(h^{m-\mu-d(1/2-1/q)_+})$ in $W^\mu_q$, and for targets of fractional order $\tau$ outside the native space the rate is $h^{\tau-\mu-d(1/2-1/q)_+}\rho^{m-\tau}$.
  • Bernstein-type inequalities give the complete inverse theorem: for nested quasi-uniform point sets refining geometrically, $L^2$ approximation at rate $h^\tau$ forces the limit field to lie in $H^{\tau'}$ for every $\tau'<\tau$.
  • The minimum eigenvalue of the div-free interpolation matrix obeys $\lambda_{\min}\ge c_d q_X^{2m-d-2k+2}$, matching the observed $q_X^7$, $q_X^5$ and $q_X^3$ decays in the experiments and quantifying the stability cost of higher-order kernels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same coefficient rule could generate kernels adapted to other differential constraints encoded as Fourier projectors—for example, curl-curl or Hodge-type decompositions—since $\alpha=(d-1)I\beta-r^2\beta$ is the radial expression of a generic projector symbol.
  • The positive-definiteness requirement in dimension $d+4-2k$ means compactly supported seeds may lose definiteness at fine scales; the paper's Wendland eigenvalue experiments hint at this, and a practical safeguard would be to verify $F_{d+4-2k}\psi_m>0$ before use.
  • The stability-to-accuracy trade-off across $k$ follows the uncertainty pattern familiar from kernel methods: smallest $k$ gives the best convergence order but the fastest decay of the minimal eigenvalue, so $k$ can be chosen according to whether the application prioritizes accuracy or conditioning.
  • If the dimension-walking decay step were proven under weaker assumptions than (4.1), the same framework would immediately cover additional families of positive-definite radial kernels beyond Matérn and Wendland.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops an operator-based framework for constructing divergence-free and curl-free matrix-valued kernels of the form K(x,y)=α(r)I+β(r)(x−y)(x−y)^T, taking β=D^k ψ_m for an integer k. The main results are: a necessary and sufficient ODE condition for the div-free constraint (Theorem 3.1), an integral representation of α (Corollary 3.3), a Fourier symbol computation (Lemma 3.6), a native-space identification with vector-valued Sobolev spaces (Theorem 4.1), and direct/inverse/stability estimates for the corresponding interpolation problem (Theorems 4.3–4.11). Numerical experiments on Matérn and Wendland kernels compare k=0,1,2 and report convergence orders and eigenvalue decays consistent with the claimed rates.

Significance. The construction is attractive: it relaxes the smoothness requirements of the classical potential-based approach, gives a simple algebraic characterization of div/curl-free kernels, and extends Bernstein and inverse theorems to matrix-valued kernels. The paper is also honest in its numerics: shape parameters are fixed, the measured convergence orders are not fitted, and the eigenvalue rates match the predicted exponents. However, the central native-space identification is proved only under a dimension-walking step that does not follow from the stated assumption (4.1), and the direct error estimates are stated only for k=1 while the numerical section claims rates for k=0 and k=2 as 'theoretical'. If these gaps are repaired by adding an explicit structural assumption and stating the general-k theorems, the paper would be a solid contribution to kernel-based vector-field approximation.

major comments (3)
  1. [Section 4, Theorem 4.1 (p. 11, Eq. (4.4))] The dimension-walking step in the proof of Theorem 4.1 is not justified. The text asserts that from F_d ψ_m(ω) ≍ (1+ω^2)^{-m} it follows that F_{d+4−2k} ψ_m(ω) ≍ (1+ω^2)^{-(m+2−k)}. The exact radial identity is F_{d+2} ψ_m(ω) = −ω^{-1} d/dω F_d ψ_m(ω) (up to normalization). From F_d ψ_m ≍ (1+ω^2)^{-m} alone one cannot control the derivative: a small oscillatory perturbation of F_d ψ_m would preserve (4.1) while making the derivative, and hence F_{d+2} ψ_m, much larger. Positive definiteness on R^{d+4−2k} only gives nonnegativity of the higher-dimensional symbol, not the required two-sided decay. Since the norm equivalence (4.4) is the basis for Theorems 4.3, 4.6, 4.7, and 4.9–4.11, all Sobolev rates in Section 4 currently hang on this unproved step. Please add an explicit structural assumption (for example, monotonicity of F_d ψ_m or a direct decay assumption on F_{d+4−2k} ψ_m) and prove the dimension-walking lemma, or restrict the statements to kernels for which the identity is known to hold.
  2. [Section 4.2, Assumption 4.4 / Theorem 4.6 / Table 5.1] The direct error estimates are stated and proved only for the case k=1: Assumption 4.4 fixes β_div = Dψ_m, and Theorem 4.6 is formulated under this assumption. However, the numerical section (Table 5.1 and Figures 5.1–5.3) reports convergence orders for K^(0) (β=ψ_m), K^(1) (β=Dψ_m), and K^(2) (β=D^2ψ_m), and labels O(h^{4.5}), O(h^{3.5}), O(h^{2.5}) as 'theoretical rates'. These rates correspond to replacing m by m+1−k in Theorem 4.6, a statement that is not proved or even stated. The sentence 'The analysis for other kernel constructions is analogous' is not a substitute for a theorem. Please state and prove the general-k version of the direct estimates, or restrict the numerical claims to the k=1 case that is actually covered.
  3. [Section 4.3, Lemma 4.8] Lemma 4.8 asserts the existence of a band-limited interpolant f_{σ,u,τ} with bandwidth σ=O(q_X^{-1}) satisfying (4.9)–(4.10), but no construction is provided. The proof refers to 'the construction of the band-limited interpolant (see, for example, [8])' without identifying the result, and the claimed stability estimate ∥f_σ∥_{\tilde H^τ(R^d)} ≤ C∥u∥_{H^τ(Ω)} is not derived. This lemma is load-bearing for Theorems 4.9–4.11, so the inverse estimates depend on it. Please provide the explicit construction (e.g., via a truncated Fourier projection or a suitable convolution kernel) and prove (4.10a)–(4.10b) in detail, or give a precise reference and verify that its hypotheses apply to the matrix-valued divergence-free setting.
minor comments (4)
  1. [Section 4, Lemma 4.5 and Lemma 4.8] The text says 'Let Ω satisfy Theorem4.4' but the relevant object is Assumption 4.4; the cross-reference should be corrected.
  2. [Section 5.1, Figure 5.4] The negative eigenvalues for Wendland kernels ϕ_{3,2} and ϕ_{5,2} are excluded from the figure; this should be stated in the caption and interpreted as a failure of positive definiteness in R^6, not merely as a numerical artifact.
  3. [Section 1, reference [18]] The text cites 'Micheli [18]', but the bibliography entry is by Micheli and Glaunes; the in-text citation should match the reference.
  4. [Section 2, Definition 2.3] In Definition 2.3(ii), the function ϕ is said to be in C^2(R), but the operator D is then used on [0,∞); the statement should explicitly say that ϕ is an even C^2 function on R whose restriction to [0,∞) is used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the dimension-walking step in Theorem 4.1 is a proof gap, not a circularity, and the numerical convergence and stability rates are genuine predictions.

full rationale

No load-bearing claim reduces by construction to its own input. The Fourier-symbol chain in Lemma 3.6 and Theorem 3.9 computes K̂(ξ)=ω²F_{d+4-2k}ψ_m(ω)I and defines the native-space norm directly from that multiplier. Theorem 4.1's norm equivalence N_K ≅ \tilde H^{m+1-k} is then a substitution of the shifted decay F_{d+4-2k}ψ_m(ω) ≍ (1+ω²)^{-(m+2-k)} into the norm integral. The proof asserts this shifted decay follows from (4.1) "by half the difference in dimensions," but Lemma 2.4 only connects F_d and F_{d±2} through I and D acting on possibly different functions; it does not transfer decay of the same ψ_m across dimensions. This is a genuine missing justification that would invalidate Theorems 4.3–4.11 if false, but it is not circular: the shifted decay is asserted, not assumed or fitted, and the experiments for Matérn and Wendland kernels verify independently predicted rates with fixed shape parameters. Self-citations [3] and [32] appear but are not load-bearing: Theorem 4.7's key H^τ bound is attributed to [8, Thm. 6], Lemma 4.8 relies on [8], and the inverse theorem uses [20,27,36]; [32] is contextual. No uniqueness theorem is imported from the authors, and the isotropic ansatz (1.4) is stated explicitly rather than smuggled in through a citation. The paper's central derivation is therefore self-contained apart from a non-circular proof gap in the dimension-walk argument.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The theoretical contribution rests on standard Fourier and Sobolev machinery plus four imported external results (dimension-walking identities, fractional div-free extension, sampling inequalities, potential-based eigenvalue bound). The only unproved assumption that looks load-bearing is the shifted-dimension decay assertion. There are no fitted parameters in the theory; the experimental shape parameters are fixed by hand and do not influence the predicted rates.

assumptions (5)
  • standard math The I and D operators satisfy Lemma 2.4: D I phi = I D phi = phi and F_d phi = F_{d-2} I phi, F_d phi = F_{d+2} D phi.
    Imported from Schaback-Wu (1996) and Wendland; used in the proofs of Lemma 3.6, Theorem 3.9, and Theorem 4.1 to compute Fourier symbols and dimension-shift decay.
  • domain assumption For psi_m with F_d psi_m(omega) ~ (1+omega^2)^{-m}, the shifted-dimension transform satisfies F_{d+4-2k} psi_m(omega) ~ (1+omega^2)^{-(m+2-k)}.
    This is the key unproved step in the proof of Theorem 4.1. It holds for generalized Wendland and Matérn kernels but is not a logical consequence of (4.1) for arbitrary radial psi_m.
  • domain assumption There exists a continuous div-free fractional extension operator E_div : H^m_div(Omega) -> tilde H^m_div(R^d), Lemma 4.5.
    Cited from Wendland [35, Prop. 3.8]; needed to transfer Sobolev norms between the bounded domain and the whole space in Theorems 4.6 and 4.7.
  • domain assumption The sampling inequalities of Arcangéli et al. [2] and Le Gia et al. [16], and the stability estimate ||f - I_X f||_{H^tau(Omega)} <= C rho^{m-tau}_X ||f||_{H^tau(Omega)} from Fuselier [8, Thm. 6] are valid here.
    Invoked without derivation in the proof of Theorem 4.7; the matrix-valued setting must inherit these scalar estimates.
  • standard math Bochner's theorem for radial functions: a scalar phi is positive definite on R^d iff F_d phi is nonnegative; the matrix kernel is positive definite when its Fourier symbol is positive semidefinite and the usual distinct-point argument applies.
    Used throughout Section 3.2 to convert Fourier formulas (3.8) into positive definiteness claims such as Examples 3.7 and 3.8.

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Pith. "Pith review of Error estimates for vector field interpolation based on generalized matrix-valued kernels." pith.science (2026). https://pith.science/paper/6C6FG373

@misc{pith2026260804313,
  author       = {Pith},
  title        = {Pith review of: Error estimates for vector field interpolation based on generalized matrix-valued kernels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6C6FG373}},
  note         = {Machine review of arXiv:2608.04313}
}
read the original abstract

Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.

Figures

Figures reproduced from arXiv: 2608.04313 by the authors.

Figure 5
Figure 5. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 5.1
Figure 5.1. Discrete ℓ2 errors for the vector field u and its div-free and curl-free components u1 and u2, computed with matrix-valued kernel interpolation using Mat´ern and Wendland kernels on uniform nodes with h = {0.4, 0.2, 0.1, 0.05, 0.025} [PITH_FULL_IMAGE:figures/full_fig_p019_5_1.png] view at source ↗
Figure 5
Figure 5. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figures from the paper (8 more)
Figure 5.2
Figure 5.2. Figure 5.2: Discrete ℓ2 errors for the vector field u and its div-free and curl-free components u1 and u2, computed with matrix-valued kernel interpolation using Mat´ern and Wendland kernels on Halton nodes with N = {6, 25, 100, 400, 1600} (with h ∼ 1/ √ N). 10−2 10−1 100 10−15 …
Figure 5.3
Figure 5.3. Figure 5.3: Minimum eigenvalues of div-free interpolation matrices using Mat´ern kernel [PITH_FULL_IMAGE:figures/full_fig_p020_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Minimum eigenvalues of interpolation matrices for the div-free component [PITH_FULL_IMAGE:figures/full_fig_p021_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: Contour of the potential ψ, streamlines of the induced vector field u, and the vector field reconstructed using K(0) . uniformly in Ω [PITH_FULL_IMAGE:figures/full_fig_p021_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Pointwise approximation errors for three cases in reconstructing the div-free field [PITH_FULL_IMAGE:figures/full_fig_p022_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Target field (5.4) in the annulus: (a) streamlines of the field; (b)–(c) scalar potentials of its decomposed components. (a) Interpolated f (b) Div-free component (c) Curl-free component [PITH_FULL_IMAGE:figures/full_fig_p022_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Contour plots for matrix-valued kernel interpolation of [PITH_FULL_IMAGE:figures/full_fig_p022_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: Pointwise approximation errors for three cases in approximating the combined field [PITH_FULL_IMAGE:figures/full_fig_p023_5_9.png]

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Reference graph

Works this paper leans on

37 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [8]

    Fuselier

    E.J. Fuselier. Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants.Math. Comput., 77(263):1407–1423, 2008

    work page 2008

  2. [1]

    Amodei and M.N

    L. Amodei and M.N. Benbourhim. A vector spline approximation.J. Approx. Theory, 67(1):51–79, 1991

    work page 1991

  3. [2]

    Arcang´ eli, M.C

    R. Arcang´ eli, M.C. L´ opez de Silanes, and J.J. Torrens. An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing.Numer. Math., 107(2):181–211, 2007

    work page 2007

  4. [3]

    Avesani, L

    S. Avesani, L. Ling, F. Marchetti, and T. Wenzel. Sobolev algorithm for local smoothness analysis (SALSA) via sharp direct and inverse statements, arXiv:2512.17377, 2025

    arXiv 2025

  5. [4]

    D.S. Balsara. Divergence-free adaptive mesh refinement for magnetohydrodynamics.J. Comput. Phys., 174(2):614–648, 2001

    work page 2001

  6. [5]

    Farrell, K

    P. Farrell, K. Gillow, and H. Wendland. Multilevel interpolation of divergence-free vector fields.IMA J. Numer. Anal., 37(1):332–353, 2017

    work page 2017

  7. [6]

    Fasshauer.Meshfree approximation methods with MATLAB, volume 6

    G.E. Fasshauer.Meshfree approximation methods with MATLAB, volume 6. World Scientific, 2007

    work page 2007

  8. [7]

    Fuselier

    E.J. Fuselier. Improved stability estimates and a characterization of the native space for matrix-valued rbfs.Adv. Comput. Math., 29(3):269–290, 2008

    work page 2008

Show all 37 references
  1. [9]

    Fuselier, V

    E.J. Fuselier, V. Shankar, and G.B. Wright. A high-order radial basis function (RBF) Leray projection method for the solution of the incompressible unsteady Stokes equations.Comput. Fluids, 128:41–52, 2016

    2016

  2. [10]

    Fuselier and G.B

    E.J. Fuselier and G.B. Wright. Stability and error estimates for vector field interpolation and decomposition on the sphere with RBFs.SIAM J. Numer. Anal., 47(5):3213–3239, 2009

    2009

  3. [11]

    Ganesh, Q.T

    M. Ganesh, Q.T. Le Gia, and I.H. Sloan. A pseudospectral quadrature method for Navier-Stokes equations on rotating spheres.Math. Comput., 80(275):1397–1430, 2011

    2011

  4. [12]

    Guzm´ an and M

    J. Guzm´ an and M. Neilan. Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput., 83(285):15–36, 2014

    2014

  5. [13]

    Handscomb

    D. Handscomb. Local recovery of a solenoidal vector field by an extension of the thin-plate spline technique. Numer. Algorithms, 5(2):121–129, 1993

    1993

  6. [14]

    Keim and H

    C. Keim and H. Wendland. A high-order, analytically divergence-free approximation method for the time-dependent Stokes problem.SIAM J. Numer. Anal., 54(2):1288–1312, 2016

    2016

  7. [15]

    K¨ unemund, F.J

    J. K¨ unemund, F.J. Narcowich, J.D. Ward, and H. Wendland. A high-order meshless Galerkin method for semilinear parabolic equations on spheres.Numer. Math., 142(2):383–419, 2019

    2019

  8. [16]

    Le Gia, F.J

    Q.T. Le Gia, F.J. Narcowich, J.D. Ward, and H. Wendland. Continuous and discrete least-squares ap- proximation by radial basis functions on spheres.J. Approx. Theory, 143(1):124–133, 2006

    2006

  9. [17]

    Lowitzsch

    S. Lowitzsch. Matrix-valued radial basis functions: stability estimates and applications.Adv. Comput. Math., 23(3):299–315, 2005

    2005

  10. [18]

    Micheli and J.A

    M. Micheli and J.A. Glaunes. Matrix-valued kernels for shape deformation analysis.Geom. Imaging Comput., 1(1):57–139, 2014

    2014

  11. [19]

    Narcowich, S.T

    F.J. Narcowich, S.T. Rowe, and J.D. Ward. A novel Galerkin method for solving PDEs on the sphere using highly localized kernel bases.Math. Comput., 86(303):197–231, 2017

    2017

  12. [20]

    Narcowich, X.P

    F.J. Narcowich, X.P. Sun, J.D. Ward, and H. Wendland. Direct and inverse Sobolev error estimates for 24Z. SUN, L. DONG AND L. LING scattered data interpolation via spherical basis functions.Found. Comput. Math., 7(3):369–390, 2007

    2007

  13. [21]

    Narcowich and J.D

    F.J. Narcowich and J.D. Ward. Generalized Hermite interpolation via matrix-valued conditionally positive definite functions.Math. Comput., 63(208):661–687, 1994

    1994

  14. [22]

    Narcowich, J.D

    F.J. Narcowich, J.D. Ward, and H. Wendland. Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting.Math. Comput., 74(250):743–763, 2005

    2005

  15. [23]

    Narcowich, J.D

    F.J. Narcowich, J.D. Ward, and H. Wendland. Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions.Constr. Approx., 24(2):175–186, 2006

    2006

  16. [24]

    Neilan and B

    M. Neilan and B. Otus. Divergence-free Scott-Vogelius elements on curved domains.SIAM J. Numer. Anal., 59(2):1090–1116, 2021

    2021

  17. [25]

    Reisert and H

    M. Reisert and H. Burkhardt. Learning equivariant functions with matrix valued kernels.J. Mach. Learn. Res., 8:385–408, 2007

    2007

  18. [26]

    Schaback

    R. Schaback. Error estimates and condition numbers for radial basis function interpolation.Adv. Comput. Math., 3(3):251–264, 1995

    1995

  19. [27]

    Schaback and H

    R. Schaback and H. Wendland. Inverse and saturation theorems for radial basis function interpolation. Math. Comput., 71(238):669–681, 2002

    2002

  20. [28]

    Schaback and Z.M

    R. Schaback and Z.M. Wu. Operators on radial functions.J. Comput. Appl. Math., 73(1-2):257–270, 1996

    1996

  21. [29]

    Schr¨ ader and H

    D. Schr¨ ader and H. Wendland. A high-order, analytically divergence-free discretization method for Darcy’s problem.Math. Comput., 80(273):263–277, 2011

    2011

  22. [30]

    Sun and L

    Z.J. Sun and L. Ling. A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations.SIAM J. Sci. Comput., 44(4):A2789–2807, 2022

    2022

  23. [31]

    Sun and L

    Z.J. Sun and L. Ling. A high-order meshless linearly implicit energy-preserving method for nonlinear wave equations on Riemannian manifolds.SIAM J. Sci. Comput., 46(6):A3779–A3802, 2024

    2024

  24. [32]

    Sun and L

    Z.J. Sun and L. Ling. Inverse inequalities for kernel-based approximation on bounded domains and rie- mannian manifolds. arXiv:2508.05376, 2025

    2025 arXiv

  25. [33]

    Wendland

    H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.Adv. Comput. Math., 4(1):389–396, 1995

    1995

  26. [34]

    Wendland.Scattered data approximation, volume 17

    H. Wendland.Scattered data approximation, volume 17. Cambridge university press, 2004

    2004

  27. [35]

    Wendland

    H. Wendland. Divergence-free kernel methods for approximating the Stokes problem.SIAM J. Numer. Anal., 47(4):3158–3179, 2009

    2009

  28. [36]

    T. Wenzel. Sharp inverse statements for kernel interpolation.Math. Comput., 95:1389–1413, 2026

    2026

  29. [37]

    Z.M. Wu. Compactly supported positive definite radial functions.Adv. Comput. Math., 4(1):283–292, 1995

    1995

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