{"id":"79cb3704-0c96-4be8-a5f1-18b27a106677","arxiv_id":"1908.01178","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Rank-1 lattice reconstruction is extended from periodic Fourier functions to nonperiodic cosine and Chebyshev functions, with exact non-aliasing conditions, smaller auxiliary index sets, and fast cosine-transform recovery.","lead":"This paper designs lattice-based sampling methods that exactly recover cosine and Chebyshev series coefficients of nonperiodic functions from a modest number of carefully chosen samples. It transfers known Fourier lattice results through tent and cosine transforms, and adds faster component-by-component construction algorithms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Plan-C exact reconstruction and CBC claims appear sound, but Theorem 27's stability bound is invalid: condition (25) does not imply V*WV=D, and an explicit n=5 example violates the stated rho_c bound.","rationale":"I read the strongest claim as the exact reconstruction characterization for cosine and Chebyshev spaces via plans A, B, and C, together with the CBC guarantees. The reader's weakest_assumption, the equal-weight character property, is an explicit premise of the paper rather than a hidden flaw; I do not see a gap in the necessity or sufficiency arguments that use it. The exact reconstruction part of the paper is mathematically sound as far as I can verify: the transfers through the tent and cosine transforms are consistent, the auxiliary sets for plan B are correctly identified modulo sign symmetry, and the CBC induction proofs in Theorems 23 and 26 cover the claimed regimes. The most load-bearing concern I find is in the approximation section. For plan C, the stability analysis assumes V*WV = D with D = diag(2^{|k|0-1}), but the plan-C non-aliasing condition (25) permits exactly the self-aliasing that makes Q_n(v_k^2) exceed d_k. A concrete n = 5, d = 2 example with Lambda = {(1,1)} and z = (1,4) satisfies all plan-C and CBC hypotheses yet violates the stated stability bound by a factor of two. This invalidates Theorem 27 as stated, though it does not affect the exact reconstruction theorems. The reader's conditional verdict was driven by an unreported numerical remark in Section 6.3; my concern is more substantive and calls for a revision of the stability constant proof. I would keep the verdict CONDITIONAL rather than REJECT because the exact reconstruction and CBC contributions remain valid and the stability fix appears localized: replacing d_k in D by the actual diagonal Q_n(v_k v_k) gives a corrected rho_c that is at most twice the stated value. I agree with the reader that the unreported numerical experiments should be removed or substantiated, but the stability gap is the more important issue to resolve.","tokens_in":32359,"tokens_out":45882,"duration_ms":478087,"concrete_test":"Evaluate the plan-C example d = 2, Lambda = {(1,1)}, n = 5, z = (1,4). Compute the lattice points i z / 5 mod 1 and v_k(x) = 2 cos(2pi(x1+x2)); since v_k(t_i) = 2 for all i, one obtains Q_n(v_k^2) = 4 while the paper's d_k = 2. Then set epsilon = (1,...,1) and compute ||C^{-1}V*W epsilon||_2^2 = 1, against the claimed bound rho_c (1/n) sum |epsilon_i|^2 = 1/2. If this violation reproduces, it settles that (42) is not a consequence of the plan-C condition (25), and that the stability constant in Theorem 27 must be corrected.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The exact reconstruction characterizations (Lemmas 15, 16, 20, 21) and the CBC existence theorems (23, 26) are internally coherent: the no-cancellation arguments rely only on the equal-weight character property (9), which is explicit in the setting. The load-bearing gap is in the approximation analysis, Sections 6.1–6.4. The derivation of the stability bound for plan C uses V*WV = D with d_k = 2^{|k|0-1}. However, Lemma 16's condition (25) does not enforce this. Since v_k = 2^{|k|0/2} cos(2pi k·x), we have v_k^2 = 2^{|k|0-1}(1 + cos(4pi k·x)); if 2k·z ≡ 0 mod n, then Q_n(v_k^2) = 2 d_k, not d_k. This is exactly the self-aliasing that plan C permits: the sign flip sigma = -id satisfies sigma(k)·z ≡ k·z, so c_k counts it. Concrete example: d = 2, Lambda = {(1,1)}, n = 5, z = (1,4). Here #Lambda #M(Lambda) = 4 < 5 and 2 max(Lambda) = 2 < 5, condition (25) is vacuous, c_k = 2, d_k = 2, but Q_n(v_k^2) = 4. The stated rho_c,k = d_k/c_k^2 = 1/2, while the actual amplification factor Q_n(v_k^2)/c_k^2 = 1. With epsilon = (1,...,1), the exact perturbation error is ||C^{-1}V*W epsilon||_2^2 = 1, violating the claimed bound rho_c (1/n) sum |epsilon_i|^2 = 1/2. Thus the proof of Theorem 27 with rho_c from (44) is not valid as written; it can be repaired by using the actual diagonal Q_n(v_k v_k) in D, which changes rho_c by at most a factor of 2, but the current statement is false.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops rank-1 lattice rules for exact integration, exact reconstruction, and approximate reconstruction of functions with finite or infinite series expansions in the Fourier, half-period cosine, and Chebyshev bases. The main mechanism is the tent transform (connecting cosine space to Fourier space) and the cosine transform (connecting Chebyshev space to cosine space), combined with the equal-weight rank-1 lattice character property. The paper states necessary and sufficient conditions for integral exactness (Lemmas 10 and 18) and for exact coefficient reconstruction under three plans: plan A using the auxiliary set M(Λ)⊕M(Λ), plan B using Λ⊕M(Λ) with a bi-orthonormal basis, and plan C allowing self-aliasing and correcting by constants c_k, with an auxiliary size #Λ #M(Λ) for the CBC construction (Lemmas 11, 15, 16 and Chebyshev analogues 19–21). It provides CBC existence proofs (Theorems 23 and 26), algorithmic strategies including smart lookups and mixed constructions, FFT/DCT coefficient recovery, and an approximation analysis culminating in Theorem 27 with stability constants. The exact reconstruction characterizations and the CBC proofs are the core contribution; the approximation and stability section contains a flaw in the plan-C stability estimate.","tokens_in":32708,"tokens_out":16013,"duration_ms":156938,"significance":"If the core characterization and CBC results are correct, the paper makes a solid contribution: it reduces the auxiliary index set for nonperiodic reconstruction from M(Λ)⊕M(Λ) to Λ⊕M(Λ) for plan B and gives a certified CBC construction for the relaxed plan C, with necessary and sufficient conditions that are non-circular and depend only on the explicit rank-1 character property. The FFT/DCT recovery algorithms and the treatment of even and odd n for Chebyshev-type lattices are useful additions. The exactness proofs are detailed, and no fitted parameters or post-hoc exclusions appear. I find the exact-reconstruction part credible. The approximation section, however, is not currently reliable because the plan-C stability bound used in Theorem 27 is false as stated; this is a local but load-bearing error in the approximation claims.","major_comments":[{"comment":"The stability analysis is not valid for plan C as written. The proof of the bound in §6.3 uses the factorization with D = diag(d_k), d_k = 2^{|k|0-1}, which requires V*WV = D, i.e., Q_n(v_k v_k') = d_k δ_{k,k'} for all k,k' in Λ. The condition in Lemma 16, Eq. (25), guarantees the off-diagonal part, but it does not control the diagonal: v_k^2 = 2^{|k|0-1}(1 + cos(4π k·x)), so when 2k·z ≡ 0 mod n we have Q_n(v_k^2) = 2^{|k|0}, not 2^{|k|0-1}. This is exactly the self-aliasing that plan C permits, and σ = -id is counted in c_k. For a concrete counterexample, take d = 2, Λ = {(1,1)}, n = 5, z = (1,4). Then condition (25) is vacuous, c_k = 2, d_k = 2, but Q_n(v_k^2) = 4. With ε = (1,...,1), the actual error ||C^{-1}V*W ε||_2^2 is 1, whereas the claimed bound with ρ^c_k = d_k/c_k^2 = 1/2 gives 1/2. Thus the identity V*WV = D displayed in §6.3 is false for plan C, and Theorem 27 with ρ^c from Eq. (44) is not established. The repair is local: one should take D = diag(Q_n(v_k v_k)) and define ρ^c from that diagonal; since Q_n(v_k^2) ≤ 2^{|k|0} and c_k ≥ 2 whenever 2k·z ≡ 0, the corrected constant is at most a factor two larger and the qualitative downstream bounds survive. As written, the theorem is incorrect.","section":"§6.3 and Theorem 27, with Lemmas 16/21"}],"minor_comments":[{"comment":"The sentence 'numerical experiments show that not much improvement can be obtained' refers to experiments that are not reported in the manuscript; either include the experiments or remove the claim.","section":"§6.3"},{"comment":"The proof delegates the full-projection case to [4,9,10]; please state exactly which of these references proves the result for arbitrary A with the full projection, since the later CBC applications use this case and the reader should not have to infer it.","section":"Theorem 23"},{"comment":"The notation 10∈Λ should be the indicator 1_{0∈Λ}; as typeset it is easy to misread as the number ten.","section":"Eq. (44)"},{"comment":"The title of Reference [26] contains a typo: 'arbitratry' should be 'arbitrary'.","section":"Reference [26]"},{"comment":"The phrase 'for sufficiently large n' could be made precise by quoting the CBC requirements, for example n > max{#(Λ⊕M(Λ)), 2 max(Λ)} for plan B and n > max{#Λ #M(Λ), 2 max(Λ)} for plan C.","section":"Theorem 27"}],"recommendation":"major_revision","confidential_remarks":"The exact-reconstruction part is the main contribution and appears sound; the stability flaw is confined to Section 6 and can be repaired by replacing D with the actual diagonal Q_n(v_k v_k). Provided the authors correct Theorem 27 and the surrounding statements, the paper is within reach. I would not require new numerical experiments for the theoretical claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is good. The exact reconstruction characterizations for cosine and Chebyshev spaces — the modular conditions in Lemmas 15, 16, 20, 21 — are proved cleanly from the rank-1 lattice character property, and the CBC construction theorems 23 and 26 give explicit cardinality bounds. The smaller auxiliary index sets (Λ ⊕ M(Λ) for plan B, and the relaxed self-aliasing for plan C) are genuinely new, as are the odd-n Chebyshev treatment via FFT/DCT and the mixed brute-force/elimination CBC strategy. The paper is also honest about prior work: the even-n plan C Chebyshev case is credited to Potts and Volkmer, and part of Lemma 11 to [29].\n\nThe soft spot is in the approximation analysis, not the reconstruction core. Section 6.3/6.4 claims a stability bound in Theorem 27 using V*WV = D with d_k = 2^{|k|0-1}. For plan C, condition (25) does not imply V*WV = D. Because self-aliasing is allowed, 2k·z can be 0 mod n, and then Q_n(v_k^2) = 2 d_k, not d_k. The paper's condition (42) is simply not established for plan C. A concrete counterexample: d=2, Λ={(1,1)}, n=5, z=(1,4) satisfies all the CBC hypotheses, c_k = 2, d_k = 2, but Q_n(v_k^2)=4. For ε=(1,...,1), the exact perturbation error is 1, while the claimed bound gives 1/2. So Theorem 27 is false as stated. The repair is straightforward — use the actual diagonal Q_n(v_k v_k) in D, which changes ρ_c by at most a factor of 2 — but the current statement needs revision.\n\nTwo smaller points. The full-projection branch of Theorem 23 is cited to [4,9,10] rather than reproved; that is acceptable, but it makes the proof less self-contained than the rest. And Section 6.3 mentions numerical experiments on plan C self-aliasing that are not reported anywhere; the remark should be substantiated or deleted.\n\nOverall: the exact reconstruction and CBC material is a solid subfield contribution. The stability gap is a real flaw in the approximation section, but it is localized and repairable. This paper deserves a serious referee; I would recommend acceptance after the authors fix Theorem 27 and its proof. If you work on quasi-Monte Carlo or polynomial reconstruction, this is worth reading and citing for the reconstruction results.","headline":"The exact reconstruction and CBC construction results are sound and genuinely new, but the plan C stability bound in Theorem 27 is false as stated and needs a localized repair.","tokens_in":33370,"tokens_out":5239,"would_cite":true,"duration_ms":48074,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A10","42A10","41A63","42B05","65D30","65D32","65D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rank-1 lattices exactly rebuild finite cosine and Chebyshev series.","keywords":["rank-1 lattices","exact function reconstruction","cosine series","Chebyshev series","tent transform","component-by-component construction","quasi-Monte Carlo integration","discrete least squares"],"falsifier":"For a small index set such as $\\Lambda=\\{(0,0),(1,0),(0,1)\\}$ and a modulus like $n=5$, enumerate all generating vectors $z\\in\\mathbb{Z}_5^2$ and compare the plan-B condition $\\sigma(k')\\cdot z\\not\\equiv k\\cdot z\\pmod 5$ for all $k\\neq k'$ with exact reconstruction of all cosine coefficients computed by formula (18). A single $z$ that reconstructs exactly while some nonzero $h\\in\\Lambda\\oplus M(\\Lambda)$ satisfies $h\\cdot z\\equiv 0\\pmod 5$ would refute Lemma 15.","tokens_in":32056,"feed_emoji":"📐","tokens_out":10472,"duration_ms":94312,"temperature":0.7,"pith_summary":"This paper establishes exact integration and coefficient reconstruction for nonperiodic functions using rank-1 lattices, the same point sets already known to integrate and reconstruct periodic Fourier series exactly. The transfer works by composing the lattice points with the tent transform, which rewrites half-period cosine basis functions as averages of exponential basis functions, and then, for Chebyshev series, with the cosine transform. The central result is a necessary-and-sufficient modular condition: a tent-transformed lattice reconstructs all cosine (or Chebyshev) coefficients on a finite index set $\\Lambda$ exactly if and only if no nonzero element of the auxiliary set $\\Lambda \\oplus M(\\Lambda)$ lies in the dual lattice, with a relaxed self-aliasing version for plan C. A component-by-component construction is proved to find such lattice generating vectors for prime $n$ above explicit cardinality bounds. If the characterization is right, exact reconstruction in these nonperiodic spaces costs no more than an FFT or DCT and needs a smaller auxiliary index set than earlier transfer arguments suggested.","feed_headline":"Rank-1 lattices exactly rebuild finite cosine and Chebyshev series","feed_subtitle":"Tent-plus-cosine transforms bring periodic exactness to nonperiodic spaces, with FFT recovery and smaller sets.","key_machinery":"The engine is the character property of equal-weight rank-1 lattices: for every integer frequency $h$, $Q_n(e_h)$ is 1 when $h\\cdot z\\equiv 0\\pmod n$ and 0 otherwise, so exactness forces individual basis sums to vanish with no cancellation. The tent transform identity $\\varphi_k(\\phi_{\\rm tent}(x))=2^{-|k|_0/2}\\sum_{\\sigma\\in S_k} e_{\\sigma(k)}(x)$ rewrites cosine basis products as sums of such exponentials, and the bi-orthonormal pair $\\{\\varphi_k\\circ\\phi_{\\rm tent}\\}$ and $\\{\\sqrt{2^{|k|_0}}\\cos(2\\pi k\\cdot \\bullet)\\}$ lets plan B reconstruct coefficients while working with $\\Lambda\\oplus M(\\Lambda)$ instead of $M(\\Lambda)\\oplus M(\\Lambda)$. The mirrored set $M(\\Lambda)=\\{\\sigma(k): k\\in\\Lambda,\\ \\sigma\\in\\{\\pm1\\}^d\\}$ and the auxiliary sets formed from it are what the component-by-component construction must test; Theorem 23 justifies that construction from projections of these sets.","core_discovery":"The paper's central claim is that exact reconstruction in the cosine and Chebyshev settings is governed by the same character property that governs periodic Fourier reconstruction, after two changes of variables. For any finite index set $\\Lambda\\subset\\mathbb{N}_0^d$, a rank-1 lattice with generating vector $z$ reconstructs all cosine coefficients by the bi-orthonormal plan-B rule exactly when $\\sigma(k')\\cdot z \\not\\equiv k\\cdot z \\pmod n$ for all $k,k'\\in\\Lambda$ and all sign changes $\\sigma\\in S_{k'}$ with $\\sigma(k')\\neq k$, equivalently when $h\\cdot z\\not\\equiv 0 \\pmod n$ for all nonzero $h\\in \\Lambda\\oplus M(\\Lambda)$. Plan A instead requires vanishing outside $M(\\Lambda)\\oplus M(\\Lambda)$, while plan C permits self-aliasing and corrects it with the normalization factor $c_k$. The same statements hold for Chebyshev series after the cosine transform, with the cubature points given by $\\cos(2\\pi iz/n)$. Theorems 23 and 26 prove that for prime $n$ above explicit bounds a component-by-component construction always finds such a $z$, and the coefficient recovery can be computed by one-dimensional FFT or DCT.","pith_inferences":["An extension the paper leaves implicit: the same reduction from $M(\\Lambda)\\oplus M(\\Lambda)$ to $\\Lambda\\oplus M(\\Lambda)$ should apply to any basis obtained by symmetrizing a periodic basis, suggesting exact reconstruction rules for sine or wavelet-type dictionaries with analogous bi-orthonormal pairs.","A testable consequence for practitioners: for even $n$, plan C with DCT-I is known equivalent to earlier Chebyshev-lattice reconstruction, but the odd-$n$ DCT-V case is new and can be checked directly in interpolation codes that currently require even point counts.","Because the necessity arguments use the binary character property so heavily, a natural next question is which non-equal-weight quadratures fail the characterization yet still reconstruct by cancellation; finding one would mark exactly where the necessary-and-sufficient boundary sits.","The approximation analysis does not optimize the generating vector for the discrete-seminorm term; combining the reconstruction condition with a worst-case error CBC criterion is a plausible route to smaller constants in the two-term error bound."],"forward_implications":["Plan B reconstructs on $\\Lambda\\oplus M(\\Lambda)$ rather than $M(\\Lambda)\\oplus M(\\Lambda)$, so the CBC construction has a strictly smaller auxiliary set to test and a lower cost bound.","Coefficient-to-value and value-to-coefficient maps for all three plans run through a one-dimensional FFT; even $n=2m$ uses a length-$m+1$ DCT-I and odd $n=2m-1$ uses a length-$m$ DCT-V.","In the approximation setting plan A has stability constant $\\rho=1$ and achieves norm equivalence with $\\delta=0$, and for downward closed $\\Lambda$ the point-count requirement improves on the earlier Chebyshev least-squares bound.","Plan C allows self-aliasing and corrects it with $c_k$, but the numerical experiments in the paper show that self-aliasing is rare, so the stability constant barely improves over plan B.","For functions not finitely supported, Theorem 27 bounds the total $L^2$ error by the truncation error plus a discrete-seminorm term, so reconstruction exactness translates directly into an approximation guarantee."],"supporting_citations":[{"why":"Establishes CBC construction of rank-1 lattices for trigonometric exactness, the periodic starting point that all later transfer arguments extend.","marker":"[4]"},{"why":"Proves exact reconstruction of hyperbolic cross trigonometric polynomials from rank-1 lattice samples, a special case generalized here.","marker":"[9]"},{"why":"Extends Fourier reconstruction along rank-1 lattices to arbitrary index sets, the periodic result Lemma 5 restates and transfers.","marker":"[10]"},{"why":"Supplies the general arbitrary-index-set Fourier reconstruction theorem and the rank-1 lattice FFT sampling framework used for recovery.","marker":"[27]"},{"why":"Introduces tent-transformed rank-1 lattices for reconstruction and collocation of nonperiodic functions; supplies Lemma 12 and part of the plan-A if-direction.","marker":"[29]"},{"why":"Gives fast exact reconstruction of multivariate polynomials in Chebyshev form for even $n$; plan C with even $n$ and DCT-I is equivalent to this approach.","marker":"[26]"},{"why":"Supplies the Markov-type inequalities for downward closed multiindex sets used to bound mirrored-set cardinalities and CBC costs.","marker":"[23]"},{"why":"Provides the discrete least-squares collocation result that the Chebyshev approximation bounds improve for downward closed sets.","marker":"[25]"}],"fun_headline_variants":["Rank-1 lattices exactly recover cosine and Chebyshev series","Exact cosine/Chebyshev recovery with a single rank-1 lattice","Bi-orthonormal plan makes rank-1 lattices exact for nonperiodic","Rank-1 lattices exactly recover finite cosine and Chebyshev coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the lattice cubature sums $Q_n(e_h)$ take only the two values $0$ and $1$ for every frequency $h$, so that when the cubature is exact the individual exponential terms cannot cancel but must vanish one by one; with weights or non-lattice points this binary character is lost and the modular characterizations no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Rank-1 lattices exactly recover cosine and Chebyshev series","Exact cosine/Chebyshev recovery with a single rank-1 lattice","Bi-orthonormal plan makes rank-1 lattices exact for nonperiodic","Rank-1 lattices exactly recover finite cosine and Chebyshev coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":3048,"prompt_tokens":955,"completion_tokens":2093,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":2011}},"tokens_in":571,"tokens_out":2093,"duration_ms":15741,"temperature":1.0,"reasoning_tokens":2011,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:22:38.557316+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small index set such as $\\Lambda=\\{(0,0),(1,0),(0,1)\\}$ and a modulus like $n=5$, enumerate all generating vectors $z\\in\\mathbb{Z}_5^2$ and compare the plan-B condition $\\sigma(k')\\cdot z\\not\\equiv k\\cdot z\\pmod 5$ for all $k\\neq k'$ with exact reconstruction of all cosine coefficients computed by formula (18). A single $z$ that reconstructs exactly while some nonzero $h\\in\\Lambda\\oplus M(\\Lambda)$ satisfies $h\\cdot z\\equiv 0\\pmod 5$ would refute Lemma 15.","supporting_citations":[{"cited_title":"Cools, F.Y","cited_arxiv_id":null,"evidence_quote":"Establishes CBC construction of rank-1 lattices for trigonometric exactness, the periodic starting point that all later transfer arguments extend."},{"cited_title":"K¨ ammerer, Reconstructing hyperbolic cross trigono metric polynomials from sampling along rank-1 lattices, SIAM J","cited_arxiv_id":null,"evidence_quote":"Proves exact reconstruction of hyperbolic cross trigonometric polynomials from rank-1 lattice samples, a special case generalized here."},{"cited_title":"K¨ ammerer, Reconstructing multivariate trigonome tric polynomials from samples along rank-1 lattices, in G.E","cited_arxiv_id":null,"evidence_quote":"Extends Fourier reconstruction along rank-1 lattices to arbitrary index sets, the periodic result Lemma 5 restates and transfers."},{"cited_title":"Potts, T","cited_arxiv_id":null,"evidence_quote":"Supplies the general arbitrary-index-set Fourier reconstruction theorem and the rank-1 lattice FFT sampling framework used for recovery."},{"cited_title":"Suryanarayana, D","cited_arxiv_id":null,"evidence_quote":"Introduces tent-transformed rank-1 lattices for reconstruction and collocation of nonperiodic functions; supplies Lemma 12 and part of the plan-A if-direction."},{"cited_title":"Potts, T","cited_arxiv_id":null,"evidence_quote":"Gives fast exact reconstruction of multivariate polynomials in Chebyshev form for even $n$; plan C with even $n$ and DCT-I is equivalent to this approach."},{"cited_title":"Migliorati, Multivariate Markov-type and Nikolski i-type inequalities for polynomials associated with downward closed multi-index sets, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Markov-type inequalities for downward closed multiindex sets used to bound mirrored-set cardinalities and CBC costs."},{"cited_title":"Narayan, Z","cited_arxiv_id":null,"evidence_quote":"Provides the discrete least-squares collocation result that the Chebyshev approximation bounds improve for downward closed sets."}],"review_version":1}