{"id":"582403c7-9c4c-4da8-a3de-ba8984057629","arxiv_id":"2608.04313","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An operator-based construction of div-free and curl-free matrix-valued kernels with Sobolev-space native spaces, plus direct, inverse, and stability estimates.","lead":"This paper builds a more flexible family of divergence-free and curl-free kernels for interpolating vector fields from scattered data, and proves approximation and stability bounds for them. The construction lowers the smoothness required of the generating scalar function, and the analysis includes direct and inverse Sobolev estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's dimension-walking step is not justified by (4.1) alone; without it the native-space identification and all direct/inverse rates collapse.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap that I find: the dimension-walking decay assertion in Theorem 4.1 is exactly the step that connects the constructed kernel's symbol to a Sobolev weight, and every subsequent direct and inverse estimate depends on that norm equivalence. My reading confirms that the proof as written contains only the sentence about half-dimension adjustment, with no derivation from Lemma 2.4 or from the decay assumption. Using the radial identity F_{d+2}ψ = -ω^{-1}F_d'ψ makes the missing ingredient explicit: one needs a two-sided bound on the derivative of F_dψ, not just on F_dψ itself. Positive definiteness in the higher dimension is not obviously sufficient for this bound, and no such argument appears in the paper. The numerical experiments are genuine supporting evidence for the Matérn and Wendland families, but they cannot establish the general theorem stated with arbitrary ψ_m satisfying (4.1). Therefore the concern is real, the verdict stays CONDITIONAL as the reader concluded, and the authors should either supply the missing hypothesis and proof or restrict the main theorem to classes where the dimension-walk step is verified.","tokens_in":20185,"tokens_out":16360,"duration_ms":158370,"concrete_test":"Independently re-derive the shifted decay in Theorem 4.1 from Lemma 2.4 using the exact identity F_{d+2}ψ_m(ω) = -ω^{-1} d/dω F_dψ_m(ω). Check whether assumption (4.1) plus positive definiteness on R^{d+4-2k} implies |d/dω F_dψ_m(ω)| ≍ ω(1+ω²)^{-m-1}. As a stress case, set F_dψ_m(ω) = (1+ω²)^{-m}(2 + sin log(1+ω²)) for m = 4 and compute F_{d+2}ψ_m; if the ratio F_{d+2}ψ_m(ω)(1+ω²)^{m+1} is not bounded both above and below, or if such a symbol cannot be radial-positive-definite in the required higher dimension, then Theorem 4.1 lacks a stated hypothesis. If the implication is provable under an added structural condition, such as complete monotonicity of F_dψ_m, the paper must state and verify that condition for the generators used in §5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 4.1 (p. 11), the proof moves from F_dψ_m(ω) ≍ (1+ω²)^{-m} to F_{d+4-2k}ψ_m(ω) ≍ (1+ω²)^{-(m+2-k)} by saying that the decay in the shifted dimension is adjusted by half the dimension difference. Lemma 2.4 only gives F_dφ = F_{d-2}(Iφ) and F_dφ = F_{d+2}(Dφ); it does not directly relate F_dψ_m to F_{d+2}ψ_m for the same ψ_m. The exact radial identity is F_{d+2}ψ_m(ω) = -ω^{-1} d/dω F_dψ_m(ω). From F_dψ_m ≍ (1+ω²)^{-m} alone it does not follow that |F_dψ_m'(ω)| ≍ ω(1+ω²)^{-m-1}: a small positive perturbation can contribute a derivative one power slower in ω. Positive definiteness on R^{d+4-2k} gives nonnegativity of the higher-dimensional symbols and hence monotonicity of lower-dimensional ones, but not the required two-sided derivative bound. Without this bound, the norm equivalence N_K ≅ \\tilde H^{m+1-k}(R^d) in (4.4) is unproved, and Theorems 4.3, 4.6, 4.7, and 4.9–4.11 all inherit the gap. The §5 experiments for Matérn and Wendland generators match the claimed rates, but they do not validate the general ψ_m class admitted by (4.1).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20510,"tokens_out":25363,"duration_ms":214388,"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":[{"comment":"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.","section":"Section 4, Theorem 4.1 (p. 11, Eq. (4.4))"},{"comment":"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.","section":"Section 4.2, Assumption 4.4 / Theorem 4.6 / Table 5.1"},{"comment":"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.","section":"Section 4.3, Lemma 4.8"}],"minor_comments":[{"comment":"The text says 'Let Ω satisfy Theorem4.4' but the relevant object is Assumption 4.4; the cross-reference should be corrected.","section":"Section 4, Lemma 4.5 and Lemma 4.8"},{"comment":"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.","section":"Section 5.1, Figure 5.4"},{"comment":"The text cites 'Micheli [18]', but the bibliography entry is by Micheli and Glaunes; the in-text citation should match the reference.","section":"Section 1, reference [18]"},{"comment":"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.","section":"Section 2, Definition 2.3"}],"recommendation":"major_revision","confidential_remarks":"The main correctness risk is concentrated in Theorem 4.1's dimension-walking step. It is a repairable gap: adding a monotonicity or direct higher-dimensional decay assumption would restore the native-space identification, and the numerical experiments for Matérn and Wendland kernels are consistent with the claimed rates. I would not recommend rejection, but the authors must either strengthen the assumptions or weaken the claims for the general class admitted by (4.1)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper genuinely delivers a cleaner way to build div-free and curl-free matrix-valued kernels: choosing beta = D^k phi instead of differentiating a high-order potential. The k=0 and k=1 recipes (Examples 3.7, 3.8) are new and practical, and the paper works out native-space norms, direct/inverse Sobolev estimates, Bernstein inequalities, and eigenvalue bounds for this class. Second, the central native-space theorem (Theorem 4.1) has a proof gap: the dimension-walking step from F_d psi_m(omega) ≍ (1+omega^2)^{-m} to F_{d+4-2k} psi_m(omega) ≍ (1+omega^2)^{-(m+2-k)} is asserted, not derived. The stress-test note is correct—this requires a two-sided bound on the derivative F_d psi_m'(omega), which (4.1) alone doesn't give. The relation F_{d+2} psi(omega) = -omega^{-1} d/domega F_d psi(omega) shows why. This is not a fatal flaw in the applied message: Matérn and Wendland kernels satisfy the stronger condition, and the experiments confirm the rates. But the theorem as stated is too broad, and the subsequent direct/inverse theorems inherit the gap.\n\nWhat the paper does well: the Fourier symbol computation in Lemma 3.6 is clean and correct; the ODE characterization (Theorem 3.1) makes the design principle transparent; and the numerical study is honest enough to report that Wendland phi_{3,2}, phi_{5,2} give negative eigenvalues for K(0), consistent with the requirement of positive definiteness on R^{d+4}. The eigenvalue decay rates match the theory. The paper also engages the literature—Fuselier, Wendland, Lowitzsch—without overclaiming what is already there.\n\nSoft spots, in proportion: (i) Theorem 4.1 is the main gap; it needs an additional structural hypothesis—for example, explicit decay for F_{d+4-2k} psi_m, or complete monotonicity, or a derivative bound—before the norm equivalence is rigorous. (ii) The proof of Theorem 4.3 imports Fuselier's eigenvalue bound wholesale; that's acceptable but reduces the novelty of the stability part. (iii) Theorem 4.6 relies on [35, Prop. 3.8] and [35, Corol. 4.7] for extension and sampling, which is fine but means the 'first complete theory' claim needs to be stated carefully. None of these are fatal; they are addressable with modest effort.\n\nWho should read this: anyone working on meshless vector-field approximation, Stokes-type collocation, or kernel-based structure-preserving methods. The construction recipes alone justify the paper, and the error analysis is useful once the gap is patched. I'd send it to a competent referee—the gap is subtle enough that a referee will catch it, but the paper is not misleading in practice. Recommend: accept after revision, with the dimension-walking step either proved under (4.1) or restricted to a class where it holds.","headline":"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.","tokens_in":21070,"tokens_out":3816,"would_cite":true,"duration_ms":33466,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A25","41A35","65D05","65D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["matrix-valued kernels","divergence-free interpolation","curl-free kernels","native space","Sobolev error estimates","Bernstein inequality","dimension-walking operators","scattered data approximation"],"falsifier":"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.","tokens_in":19963,"feed_emoji":"🌀","tokens_out":14084,"duration_ms":121590,"temperature":0.7,"pith_summary":"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 $k=0,1,2$.","feed_headline":"Divergence-free kernels from rough seeds: error rates proven","feed_subtitle":"An operator-based construction preserves incompressibility while matching sharp Sobolev convergence rates.","key_machinery":"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.","core_discovery":"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 $c_d q_X^{2m-d-2k+2}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the I and D operators and the dimension-walking identities between radial Fourier transforms that the kernel construction is built on.","marker":"[28]"},{"why":"Provides the Bochner characterization, radial Fourier transform formalism, and native-space framework used throughout.","marker":"[34]"},{"why":"Gives the stability estimate and native-space characterization for matrix-valued RBFs invoked in Theorem 4.3.","marker":"[7]"},{"why":"Establishes the Sobolev approximation rates for div-free and curl-free RBF interpolants that Theorems 4.6 and 4.7 extend to fractional regularity.","marker":"[8]"},{"why":"Supplies the div-free extension operator and the sampling bound used to prove the direct error estimates.","marker":"[35]"},{"why":"Provides the Sobolev sampling inequalities for functions with scattered zeros used in the direct estimates.","marker":"[22]"},{"why":"Supplies the sampling inequality applied in Theorem 4.7 for targets with fractional smoothness.","marker":"[16]"},{"why":"Serves as the sharp inverse-theorem template adapted to matrix-valued kernels in Theorem 4.11.","marker":"[36]"}],"fun_headline_variants":["Div-free kernels from rough seeds: proven Sobolev rates","Matrix kernels with less smoothness, same interpolation accuracy","Fractional Sobolev error bounds for constrained kernels","Sharp inverse theorem for divergence-free kernel fits","Relaxed regularity kernels with optimal error control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Div-free kernels from rough seeds: proven Sobolev rates","Matrix kernels with less smoothness, same interpolation accuracy","Fractional Sobolev error bounds for constrained kernels","Sharp inverse theorem for divergence-free kernel fits","Relaxed regularity kernels with optimal error control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000872,"raw_usage":{"total_tokens":3864,"prompt_tokens":1120,"completion_tokens":2744,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":736,"completion_tokens_details":{"reasoning_tokens":2679}},"tokens_in":736,"tokens_out":2744,"duration_ms":22625,"temperature":1.0,"reasoning_tokens":2679,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:05:06.590188+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Schaback and Z.M","cited_arxiv_id":null,"evidence_quote":"Supplies the I and D operators and the dimension-walking identities between radial Fourier transforms that the kernel construction is built on."},{"cited_title":"Wendland.Scattered data approximation, volume 17","cited_arxiv_id":null,"evidence_quote":"Provides the Bochner characterization, radial Fourier transform formalism, and native-space framework used throughout."},{"cited_title":"Fuselier","cited_arxiv_id":null,"evidence_quote":"Gives the stability estimate and native-space characterization for matrix-valued RBFs invoked in Theorem 4.3."},{"cited_title":"Fuselier","cited_arxiv_id":null,"evidence_quote":"Establishes the Sobolev approximation rates for div-free and curl-free RBF interpolants that Theorems 4.6 and 4.7 extend to fractional regularity."},{"cited_title":"Wendland","cited_arxiv_id":null,"evidence_quote":"Supplies the div-free extension operator and the sampling bound used to prove the direct error estimates."},{"cited_title":"Narcowich, J.D","cited_arxiv_id":null,"evidence_quote":"Provides the Sobolev sampling inequalities for functions with scattered zeros used in the direct estimates."},{"cited_title":"Le Gia, F.J","cited_arxiv_id":null,"evidence_quote":"Supplies the sampling inequality applied in Theorem 4.7 for targets with fractional smoothness."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Serves as the sharp inverse-theorem template adapted to matrix-valued kernels in Theorem 4.11."}],"review_version":1}