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REVIEW 1 major objections 5 minor 36 references

Function integration, reconstruction and approximation using rank-1 lattices

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Pith's one-line read Rank-1 lattices exactly rebuild finite cosine and Chebyshev series.

desk verdict 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. read the letter →

arxiv 1908.01178 v4 pith:WTRGOMOK submitted 2019-08-03 math.NA cs.NA

Frances Y. Kuo , Giovanni Migliorati , Fabio Nobile , Dirk Nuyens This is my paper · ORCID
classification math.NAcs.NA MSC 41A1042A1041A6342B0565D3065D3265D15
keywords rank-1latticesexactfunctionreconstructioncosineseriesChebyshevtenttransformcomponent-by-componentconstructionquasi-MonteCarlointegrationdiscreteleastsquares
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

1 major / 5 minor

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.

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 (1)
  1. [§6.3 and Theorem 27, with Lemmas 16/21] 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.
minor comments (5)
  1. [§6.3] 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.
  2. [Theorem 23] 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.
  3. [Eq. (44)] The notation 10∈Λ should be the indicator 1_{0∈Λ}; as typeset it is easy to misread as the number ten.
  4. [Reference [26]] The title of Reference [26] contains a typo: 'arbitratry' should be 'arbitrary'.
  5. [Theorem 27] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the nonperiodic reconstruction conditions are derived from the rank-1 lattice character property, and the self-citations serve only as external inputs or benchmarks.

full rationale

The derivation chain is self-contained. The load-bearing exactness and reconstruction characterizations (Lemmas 10, 11, 15, 16, 20, 21) are proven from the rank-1 lattice character property (9), which gives binary values Q_n(e_h) in {0,1}; no cancellation argument is hidden or presupposed. The plan-B condition (19) is shown equivalent to (20) by the definition of M(Lambda), and plan C's c_k is a normalization computed from the chosen generating vector z, not a fitted parameter. The paper cites [4], [9], [10], [27], and [29] for the periodic starting point and tent-transform multiplicity, but those are external results used as inputs and benchmarks; they do not assert the target nonperiodic characterizations, and no conclusion is forced by an author self-citation chain. The only flagged concern, the validity of the stability bound in Theorem 27 (condition (25) may not imply V*WV = D), is a mathematical correctness issue, not circularity: it does not reduce a prediction to its input. Accordingly the circularity score is low.

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

No free parameters are fitted to data and no new physical or mathematical objects are postulated. The results are derived from standard Fourier analysis, the tent transform, the cosine transform, and the rank-1 lattice character property; the finite-support function model is stated explicitly.

assumptions (5)
  • domain assumption Finite-index support model
    Exact integration and reconstruction results are stated for F_Lambda, functions whose series are supported only on a finite index set; Section 6 handles general functions via truncation and error bounds.
  • standard math Rank-1 lattice character property
    Equation (9): Q_n(e_h) is 1 if h dot z is congruent to 0 mod n and 0 otherwise. This binary character is used in all necessity arguments to exclude cancellation.
  • standard math Tent-transform identity (14)
    phi_k(phi_tent(x)) = 2^{-|k|_0/2} sum_{sigma in S_k} e_{sigma(k)}(x). This is the bridge from cosine coefficients to Fourier coefficients.
  • standard math Cosine-transform isomorphism
    eta_k(cos(pi x')) = phi_k(x') and the measure change identify Chebyshev coefficients with cosine coefficients (Section 4.2).
  • domain assumption Prime lattice size with large enough n
    The CBC existence proofs in Theorems 23 and 26 require n prime and n exceeding explicit cardinality bounds; composite n may work in practice but is not covered by the guarantee.

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Cite this review

Pith. "Pith review of Function integration, reconstruction and approximation using rank-1 lattices." pith.science (2026). https://pith.science/paper/WTRGOMOK

@misc{pith2026190801178,
  author       = {Pith},
  title        = {Pith review of: Function integration, reconstruction and approximation using rank-1 lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTRGOMOK}},
  note         = {Machine review of arXiv:1908.01178}
}
read the original abstract

We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [29]

    Suryanarayana, D

    G. Suryanarayana, D. Nuyens, R. Cools, Reconstruction and collocation of a class of non- periodic functions by sampling along tent-transformed ran k-1 lattices, Journal of Fourier Analysis and Applications, 22(1):187–214, 2016. 33

    work page 2016

  2. [1]

    Byrenheid, L

    G. Byrenheid, L. K¨ ammerer, T. Ullrich, T. Volkmer, Tigh t error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness, Numer. Math. , 136:993–1034, 2017

    work page 2017

  3. [2]

    Chkifa, A

    A. Chkifa, A. Cohen, G. Migliorati, F. Nobile, R. Tempone , Discrete least-squares poly- nomial approximation with random evaluations – applicatio n to parametric and stochastic elliptic PDEs, ESAIM Math. Model. Numer. Anal., 49(3):815– 837, 2015

    work page 2015

  4. [3]

    Cohen, M

    A. Cohen, M. Davenport, D. Leviatan, On the stability and accuracy of least-squares ap- proximations, Found. Comput. Math., 5:819–834, 2013

    work page 2013

  5. [4]

    Cools, F.Y

    R. Cools, F.Y. Kuo, D. Nuyens, Constructing lattice rule s based on weighted degree of exactness and worst case error, Computing, 87:63–89, 2010

    work page 2010

  6. [5]

    Cools, F.Y

    R. Cools, F.Y. Kuo, D. Nuyens, G. Suryanarayana, Tent-tr ansformed lattice rules for inte- gration and approximation of multivariate non-periodic fu nctions, J. Complexity, 36:166– 181, 2016

    work page 2016

  7. [6]

    Cools, K

    R. Cools, K. Poppe, Chebyshev lattices, a unifying frame work for cubature with Chebyshev weight function, BIT Numer. Math., 51:275–288, 2011

    work page 2011

  8. [7]

    J. Dick, D. Nuyens, F. Pillichshammer, Lattice rules for nonperiodic smooth integrands, Numer. Math., 126:259–291, 2014

    work page 2014

Show all 36 references
  1. [8]

    F. J. Hickernell, Obtaining O(N −2+ǫ) convergence for lattice quadrature rules, in K. T. Fang, F. J. Hickernell, and H. Niederreiter (Eds.), Monte Ca rlo and Quasi-Monte Carlo Methods 2000, Springer, 274–289, 2002

    2000

  2. [9]

    K¨ ammerer, Reconstructing hyperbolic cross trigono metric polynomials from sampling along rank-1 lattices, SIAM J

    L. K¨ ammerer, Reconstructing hyperbolic cross trigono metric polynomials from sampling along rank-1 lattices, SIAM J. Numer. Anal., 2773–2796, 201 3

  3. [10]

    K¨ ammerer, Reconstructing multivariate trigonome tric polynomials from samples along rank-1 lattices, in G.E

    L. K¨ ammerer, Reconstructing multivariate trigonome tric polynomials from samples along rank-1 lattices, in G.E. Fasshauer, L.L. Schumaker (Eds.), Approximation Theory XIV: San Antonio 2013, Springer, 255–271, 2014

    2013

  4. [11]

    K¨ ammerer, D

    L. K¨ ammerer, D. Potts, T. Volkmer, Approximation of mu ltivariate periodic functions by trigonometric polynomials based on rank-1 lattice samplin g, J. Complexity, 31:543–576, 2015

    2015

  5. [12]

    K¨ ammerer, S

    L. K¨ ammerer, S. Kunis, D. Potts, Interpolation lattic es for hyperbolic cross trigonometric polynomials, J. Complexity, 28:76–92, 2012. 32

    2012

  6. [13]

    K¨ ammerer, T

    L. K¨ ammerer, T. Volkmer, Approximation of multivariate periodic functions based on sam- pling along multiple rank-1 lattices, J. Approx. Theory, 24 6:1–27, 2019

    2019

  7. [14]

    F. Y. Kuo, I. H. Sloan, H. Wo´ zniakowski, Lattice rules for multivariate approximation in the worst case setting, in H. Niederreiter and D. Talay (Eds.), M onte Carlo and Quasi-Monte Carlo Methods 2004, Springer, 289–330, 2006

    2004

  8. [15]

    F. Y. Kuo, I. H. Sloan, H. Wo´ zniakowski, Lattice rule al gorithms for multivariate approxi- mation in the average case setting, J. Complexity, 24:283–3 23, 2008

    2008

  9. [16]

    F. Y. Kuo, G. W. Wasilkowski, H. Wo´ zniakowski, Multiva riate L∞ approximation in the worst case setting over reproducing kernel Hilbert spaces, J. Approx. Theory, 152:135–160, 2008

    2008

  10. [17]

    F. Y. Kuo, G. W. Wasilkowski, H. Wo´ zniakowski, On the po wer of standard information for multivariate approximation in the worst case setting, J . Approx. Theory, 158:97–125, 2009

    2009

  11. [18]

    F. Y. Kuo, G. W. Wasilkowski, H. Wo´ zniakowski, Lattice algorithms for multivariate L∞ approximation in the worst-case setting, Constr. Approx., 30:475–493, 2009

    2009

  12. [19]

    D. Li, F. J. Hickernell, Trigonometric spectral colloc ation methods on lattices, in S. Y. Cheng, C.-W. Shu, and T. Tang (Eds.), Recent Advances in Scie ntific Computing and Par- tial Differential Equations, AMS Series in Contemporary Math ematics, vol. 330, American Mathematica...

    2003

  13. [20]

    S. A. Martucci, Symmetric convolution and the discrete sine and cosine transforms, IEEE Transactions on Signal Processing, 42:1038–1051, 1994

    1994

  14. [21]

    Migliorati, F

    G. Migliorati, F. Nobile, E. von Schwerin, R.Tempone, A nalysis of discrete L2 projection on polynomial spaces with random evaluations, Found. Compu t. Math. 14:419–456, 2014

    2014

  15. [22]

    Migliorati, F

    G. Migliorati, F. Nobile, Analysis of discrete least sq uares on multivariate polynomial spaces with evaluations at low-discrepancy point sets, J. Complex ity, 31(4):517–542, 2015

    2015

  16. [23]

    Migliorati, Multivariate Markov-type and Nikolski i-type inequalities for polynomials associated with downward closed multi-index sets, J

    G. Migliorati, Multivariate Markov-type and Nikolski i-type inequalities for polynomials associated with downward closed multi-index sets, J. Appro x. Theory, 189:137–159, 2015

    2015

  17. [24]

    Munthe-Kaas, T

    H. Munthe-Kaas, T. Sørevik, Multidimensional pseudo- spectral methods on lattice grids, Appl. Numer. Math., 62:155–165, 2012

    2012

  18. [25]

    Narayan, Z

    A. Narayan, Z. Xu, T. Zhou, Multivariate discrete least -squares approximations with a new type of collocation grid, SIAM J. Sci. Comput., 36(5):A2401 –A2422, 2014

    2014

  19. [26]

    Potts, T

    D. Potts, T. Volkmer, Fast and exact reconstruction of a rbitratry multivariate algebraic polynomials in Chebyshev form, 2015 International Confere nce on Sampling Theory and Applications (SampTA), IEEE, 392–396, 2015

    2015

  20. [27]

    Potts, T

    D. Potts, T. Volkmer, Sparse high-dimensional FFT base d on rank-1 lattice sampling, Appl. Comput. Harmon. Anal. 41:713–748, 2016

    2016

  21. [28]

    I. H. Sloan, S. Joe, Lattice methods for multiple integr ation, Oxford University Press, Oxford, 1994

    1994

  22. [30]

    Suzuki, G

    Y. Suzuki, G. Suryanarayana, D. Nuyens, Strang splitti ng in combination with rank-1 and rank- r lattices for the time-dependent Schr¨ odinger equation, SI AM J. Sci. Comput., 41(6):B1254–B1283, 2019

    2019

  23. [31]

    Suzuki, D

    Y. Suzuki, D. Nuyens, Rank-1 lattices and higher-order exponential splitting for the time- dependent Schr¨ odinger equation, in B. Tuffin, and P. L’Ecuye r (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2018, Springer, 485–502, 2020

    2018

  24. [32]

    Volkmer, sparseFFTr1l software library, https://w ww-user.tu-chemnitz.de/∼tovo/software.php.en

    T. Volkmer, sparseFFTr1l software library, https://w ww-user.tu-chemnitz.de/∼tovo/software.php.en

  25. [33]

    G. W. Wasilkowski, H. Wo´ zniakowski, Weighted tensor p roduct algorithms for linear mul- tivariate problems, J. Complexity, 15:402–447, 1999

    1999

  26. [34]

    G. W. Wasilkowski, H. Wo´ zniakowski, On the power of sta ndard information for weighted approximation, Found. Comput. Math., 1:417–434, 2001

    2001

  27. [35]

    X. Y. Zeng, K. T. Leung, F. J. Hickernell, Error analysis of splines for periodic problems using lattice designs, in H. Niederreiter and D. Talay (Eds. ), Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, 501–514, 2006

    2004

  28. [36]

    X. Y. Zeng, P. Kritzer, F. J. Hickernell, Spline methods using integration lattices and digital nets, Constr. Approx., 30: 529–555, 2009. 34

    2009

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