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arxiv: 2412.08044 · v3 · submitted 2024-12-11 · 🧮 math.NA · cs.NA

Scaling Optimized Hermite Approximation Methods

Hao Hu , Haijun Yu This is my paper

Pith reviewed 2026-05-23 07:41 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Hermite functionsscaled Hermite approximationspectral methodstruncation errorL2 projectionconvergence ratesalgebraic decaygeometric convergence
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The pith

Proper scaling of Hermite functions balances spatial and frequency truncation errors to restore optimal convergence rates.

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

The paper argues that Hermite spectral methods appear inefficient only because the decay rates of the target function in the spatial and frequency domains are mismatched. It introduces an error framework that splits the L2 projection error into three parts: spatial truncation, frequency truncation, and the remaining Hermite approximation error. Setting the scaling parameter so that the first two errors are equal removes the mismatch. This choice recovers geometric convergence for one class of functions and doubles the algebraic convergence order for smooth functions whose tails decay algebraically, while also accounting for the sub-geometric rates seen in the pre-asymptotic regime.

Core claim

Finding the optimal scaling factor for Hermite functions is equivalent to equating the spatial truncation error with the frequency truncation error. With this choice, geometric convergence is recovered for a class of functions, the convergence order for smooth algebraically decaying functions is doubled, and the previously puzzling pre-asymptotic sub-geometric convergence for algebraic decay functions is fully explained by the imbalance of the two truncation terms.

What carries the argument

The error decomposition framework that expresses the L2 projection error as the sum of spatial truncation error, frequency truncation error, and Hermite spectral approximation error, with the optimal scale obtained by equating the first two terms.

If this is right

  • Geometric convergence rates are restored for functions whose spatial and frequency decays can be matched by a single scale factor.
  • The algebraic convergence order for smooth functions with power-law tails is doubled once the truncation errors are balanced.
  • The observed pre-asymptotic sub-geometric regime for algebraically decaying functions is produced exactly by the imbalance between spatial and frequency truncation.
  • Hermite spectral methods become competitive with other spectral bases once the scaling parameter is chosen according to the balancing rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition idea could be tested on other orthogonal polynomial families that admit a scaling parameter.
  • The framework supplies an explicit rule for choosing the scale when Hermite expansions are used inside time-stepping schemes for PDEs.
  • Direct numerical checks on the location of the minimal error versus the predicted balancing scale would provide immediate verification or refutation.

Load-bearing premise

The L2 projection error can be split into three separate components whose sizes can be controlled independently, so that the best scale is found simply by making the spatial and frequency truncation errors equal.

What would settle it

For a concrete test function, compute the actual L2 projection error over a range of scaling parameters and check whether the minimum occurs at the scale predicted by equating the spatial and frequency truncation errors; a clear mismatch would disprove the balancing claim.

read the original abstract

Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the $L^2$ projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the inefficiency of standard Hermite spectral methods arises from imbalance between spatial and frequency decay rates of the target function. It introduces a novel error-analysis framework for scaled Hermite approximations, illustrated on the L² projection error, which decomposes into three components—spatial truncation error, frequency truncation error, and Hermite spectral approximation error—and asserts that the optimal scaling factor is obtained by equating the first two. Applications are given showing recovery of geometric convergence for a class of functions and doubling of algebraic convergence order for smooth functions with algebraic decay; the framework is also said to explain pre-asymptotic sub-geometric behavior.

Significance. If the decomposition and balancing argument are rigorous, the work supplies a concrete, theoretically grounded procedure for choosing the scaling parameter that can restore optimal rates for Hermite methods on functions whose decay is mismatched to the standard basis, thereby addressing a long-standing practical limitation and making Hermite spectral methods competitive with other global bases for selected function classes.

major comments (2)
  1. [Abstract, §3] Abstract and §3 (error-analysis framework): the claim that the L² projection error decomposes into three independently controllable components with the optimal scale determined solely by equating spatial and frequency truncation errors is not justified. Because the scaled Hermite functions themselves depend on the scaling parameter, the orthogonal projection onto the finite scaled space (and therefore the third error term) varies with the scale; treating it as an additive, scale-independent remainder undermines the asserted equivalence between optimal scaling and balancing only the truncation errors.
  2. [§4] §4 (applications to geometric and algebraic convergence): the stated recovery of geometric convergence and doubling of algebraic order rest on the same three-term decomposition. Without an explicit bound showing that the Hermite spectral approximation error remains negligible or can be controlled uniformly once the first two terms are balanced, the convergence-rate claims lack a complete proof.
minor comments (2)
  1. [Abstract, Introduction] Notation for the scaling parameter and the scaled basis should be introduced once, early, and used consistently; several passages in the abstract and introduction repeat the same definition.
  2. [§3] The manuscript would benefit from a short table or diagram that explicitly lists the three error components, their dependence (or independence) on the scaling factor, and the quantities that are set equal to obtain the optimal scale.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The major comments raise important points about the rigor of the error decomposition and the completeness of the convergence proofs. We address each below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (error-analysis framework): the claim that the L² projection error decomposes into three independently controllable components with the optimal scale determined solely by equating spatial and frequency truncation errors is not justified. Because the scaled Hermite functions themselves depend on the scaling parameter, the orthogonal projection onto the finite scaled space (and therefore the third error term) varies with the scale; treating it as an additive, scale-independent remainder undermines the asserted equivalence between optimal scaling and balancing only the truncation errors.

    Authors: We appreciate the referee's observation that the scaled basis depends on the parameter, so the projection (and thus the third term) is not independent of scale. In our framework the decomposition begins by applying spatial and frequency truncations to the target function before projecting the resulting compactly supported function onto the scaled space; the third term is therefore the projection error of this truncated function. While the manuscript treats the terms as separately controllable for the purpose of identifying the balancing scale, the referee is correct that a uniform bound on the scale-dependent projection error is needed to justify the equivalence. We will revise §3 to include such a bound, showing that under the balanced scaling the third term is of strictly higher order than the balanced truncation errors for the function classes considered. revision: yes

  2. Referee: [§4] §4 (applications to geometric and algebraic convergence): the stated recovery of geometric convergence and doubling of algebraic order rest on the same three-term decomposition. Without an explicit bound showing that the Hermite spectral approximation error remains negligible or can be controlled uniformly once the first two terms are balanced, the convergence-rate claims lack a complete proof.

    Authors: The referee correctly notes that the rate claims in §4 presuppose control of the third term once the first two are balanced. The applications derive the rates by equating the truncation errors and then argue that the projection error of the truncated function is asymptotically smaller; however, an explicit uniform bound is not supplied. We will add this bound in the revised §4 (and the supporting analysis in §3), confirming that the third term does not alter the leading geometric or doubled algebraic rates. This will complete the proofs for the stated convergence results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on proposed truncation framework without reduction to inputs by construction.

full rationale

The paper introduces a novel error analysis framework decomposing L2 projection error into spatial truncation, frequency truncation, and Hermite spectral approximation errors, then equates the first two for optimal scaling. No quoted steps reduce predictions to fitted parameters from target data, self-citations that are load-bearing, or self-definitional equivalences. The framework is presented as derived from standard truncation analysis rather than circularly from its own outputs. This is a self-contained proposal against external benchmarks, consistent with a low circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard properties of orthogonal polynomials and L2 spaces; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Hermite functions form a complete orthogonal basis for the weighted L2 space with Gaussian weight.
    Invoked implicitly when discussing spectral approximation and truncation errors.

pith-pipeline@v0.9.0 · 5761 in / 1223 out tokens · 24650 ms · 2026-05-23T07:41:21.688645+00:00 · methodology

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Convergence theory for Hermite approximations under adaptive coordinate transformations

    math.NA 2026-04 unverdicted novelty 6.0

    Hermite approximations composed with adaptive transformations are equivalent to standard Hermite approximation of the pullback function, yielding error bounds controlled by the regularity of that pullback and enabling...

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Works this paper leans on

55 extracted references · 55 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aguirre and J

    J. Aguirre and J. Rivas , Hermite pseudospectral approximations. An error estimate , Journal of Mathe- matical Analysis and Applications, 304 (2005), pp. 189–197

    work page 2005

  2. [2]

    Babuˇska, F

    I. Babuˇska, F. Nobile, and R. Tempone , A stochastic collocation method for elliptic partial differential equations with random input data , SIAM J. Numer. Anal., 45 (2007), pp. 1005–1034, https://doi.org/ 10.1137/050645142

    work page doi:10.1137/050645142 2007

  3. [3]

    Bao and J

    W. Bao and J. Shen , A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose– Einstein condensates, SIAM J. Sci. Comput., 26 (2005), https://doi.org/10/dj68cc

    work page 2005

  4. [4]

    Barrett, Convergence properties of Gaussian quadrature formulae , The Computer Journal, 3 (1961), pp

    W. Barrett, Convergence properties of Gaussian quadrature formulae , The Computer Journal, 3 (1961), pp. 272–277

    work page 1961

  5. [5]

    J. P. Boyd, The rate of convergence of Hermite function series , Mathematics of Computation, 35 (1980), pp. 1309–1316, https://doi.org/10/bqfgdp

    work page 1980

  6. [6]

    J. P. Boyd, Asymptotic coefficients of Hermite function series, Journal of Computational Physics, 54 (1984), pp. 382–410, https://doi.org/10/dvwh5h

    work page 1984

  7. [7]

    J. P. Boyd , The Fourier transform of the quartic Gaussian exp(−x4): Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to exp(−ax2n), Applied Math- ematics and Computation, 241 (2014), pp. 75–87

    work page 2014

  8. [8]

    Cai and R

    Z. Cai and R. Li , Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation , SIAM J. Sci. Comput., 32 (2010), https://doi.org/10.1137/100785466

    work page doi:10.1137/100785466 2010

  9. [9]

    R. H. Cameron and W. T. Martin , The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annals of Mathematics, 48 (1947), pp. 385–392, https://doi.org/10.2307/ 1969178

    work page 1947

  10. [10]

    Chen, Y.-W

    J. Chen, Y.-W. Chang, and Y.-C. Huang , Analytical placement considering the electron-beam fogging effect, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 40 (2021), pp. 560–573, https://doi.org/ 10/gssf7n

    work page 2021

  11. [11]

    T. S. Chihara, Generalized Hermite polynomials, Purdue University, 1955

    work page 1955

  12. [12]

    T. Chou, S. Shao, and M. Xia , Adaptive hermite spectral methods in unbounded domains , Applied Nu- merical Mathematics, 183 (2023), pp. 201–220

    work page 2023

  13. [13]

    P. J. Davis and P. Rabinowitz , Methods of Numerical Integration, Courier Corporation, 2007

    work page 2007

  14. [14]

    Duoandikoetxea, Fourier Analysis, vol

    J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical SCALING OPTIMIZED HERMITE APPROXIMATION METHODS 21 Society, Providence, Rhode Island, Dec. 2000, https://doi.org/10.1090/gsm/029

    work page doi:10.1090/gsm/029 2000

  15. [15]

    Funaro and G

    D. Funaro and G. Manzini , Stability and conservation properties of Hermite-based approximations of the Vlasov-Poisson system, J. Sci. Comput., 88 (2021), p. 29, https://doi.org/10/gsskwz

    work page 2021

  16. [16]

    Gautschi and R

    W. Gautschi and R. S. Varga , Error bounds for Gaussian quadrature of analytic functions , SIAM J. Numer. Anal., 20 (1983), pp. 1170–1186, https://doi.org/10/b7t6zd

    work page 1983

  17. [17]

    Gibelli and B

    L. Gibelli and B. D. Shizgal , Spectral convergence of the Hermite basis function solution of the Vlasov equation: The free-streaming term , Journal of Computational Physics, 219 (2006), pp. 477–488, https: //doi.org/10.1016/j.jcp.2006.06.017

    work page doi:10.1016/j.jcp.2006.06.017 2006

  18. [18]

    T. Goda, Y. Kazashi, and K. Tanaka , How sharp are error bounds?–lower bounds on quadrature worst- case errors for analytic functions– , SIAM Journal on Numerical Analysis, 62 (2024), pp. 2370–2392

    work page 2024

  19. [19]

    Gottlieb and S

    D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications , SIAM, 1977

    work page 1977

  20. [20]

    Greengard and J

    L. Greengard and J. Strain, The fast Gauss transform, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 79–94, https://doi.org/10.1137/0912004

    work page doi:10.1137/0912004 1991

  21. [21]

    Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations , Math- ematics of Computation, 68 (1999), pp

    B.-Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations , Math- ematics of Computation, 68 (1999), pp. 1067–1078

    work page 1999

  22. [22]

    Hardy, A theorem concerning Fourier transforms, Journal of the London Mathematical Society, 1 (1933), pp

    G. Hardy, A theorem concerning Fourier transforms, Journal of the London Mathematical Society, 1 (1933), pp. 227–231

    work page 1933

  23. [23]

    Y. He, P. Li, and J. Shen , A new spectral method for numerical solution of the unbounded rough surface scattering problem, Journal of Computational Physics, 275 (2014), pp. 608–625, https://doi.org/10. 1016/j.jcp.2014.07.026

    work page 2014

  24. [24]

    Huang and H

    S. Huang and H. Yu, Improved Laguerre spectral methods with less round-off errors and better stability , J. Sci. Comput., 101,72 (2024), https://doi.org/10.1007/s10915-024-02725-9

    work page doi:10.1007/s10915-024-02725-9 2024

  25. [25]

    Kazashi, Y

    Y. Kazashi, Y. Suzuki, and T. Goda , Suboptimality of Gauss–Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness , SIAM J. Numer. Anal., 61 (2023), https: //doi.org/10.1137/22M1480276

    work page doi:10.1137/22m1480276 2023

  26. [26]

    Luo and S

    X. Luo and S. S.-T. Yau, Hermite spectral method with hyperbolic cross approximations to high-dimensional parabolic PDEs, SIAM Journal on Numerical Analysis, 51 (2013), pp. 3186–3212, https://doi.org/10. 1137/120896931

    work page 2013

  27. [27]

    H. Ma, W. Sun, and T. Tang , Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains , SIAM J. Numer. Anal., 43 (2005), pp. 58–75, https://doi.org/10/ bmh8vd

    work page 2005

  28. [28]

    Manfredi, R

    P. Manfredi, R. Trinchero, and D. V. Ginste , A perturbative stochastic Galerkin method for the uncer- tainty quantification of linear circuits , IEEE Transactions on Circuits and Systems I: Regular Papers, 67 (2020), pp. 2993–3006, https://doi.org/10.1109/TCSI.2020.2987470

    work page doi:10.1109/tcsi.2020.2987470 2020

  29. [29]

    W. C. Meecham and A. Siegel, Wiener-Hermite expansion in model turbulence at large Reynolds numbers, The Physics of Fluids, 7 (1964), pp. 1178–1190, https://doi.org/10.1063/1.1711359

    work page doi:10.1063/1.1711359 1964

  30. [30]

    Mieussens, Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries , Journal of Computational Physics, 162 (2000), pp

    L. Mieussens, Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries , Journal of Computational Physics, 162 (2000), pp. 429–466, https://doi. org/10.1006/jcph.2000.6548

    work page doi:10.1006/jcph.2000.6548 2000

  31. [31]

    Mizerov ´a and B

    H. Mizerov ´a and B. She , A conservative scheme for the Fokker–Planck equation with applications to viscoelastic polymeric fluids , Journal of Computational Physics, 374 (2018), pp. 941–953, https://doi. org/10/gfgscr

    work page 2018

  32. [32]

    Nobile, L

    F. Nobile, L. Tamellini, F. Tesei, and R. Tempone , An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient, in Sparse Grids and Applications - Stuttgart 2014, J. Garcke and D. Pfl¨ uger, eds., Lecture Notes in Computational Science and Engineering, Cham, 2016, Springer International Publishing, pp. 191–220, https://doi.or...

    work page 2014

  33. [33]

    F. W. J. Olver , ed., Nist Handbook of Mathematical Functions , Cambridge ; New York, 2010. OCLC: ocn502037224

    work page 2010

  34. [34]

    S. A. Orszag and L. R. Bissonnette, Dynamical properties of truncated Wiener-Hermite expansions, The Physics of Fluids, 10 (1967), pp. 2603–2613, https://doi.org/10.1063/1.1762082

    work page doi:10.1063/1.1762082 1967

  35. [35]

    Shan and X

    X. Shan and X. He , Discretization of the velocity space in the solution of the Boltzmann equation , Phys. Rev. Lett., 80 (1998), pp. 65–68, https://doi.org/10.1103/PhysRevLett.80.65

    work page doi:10.1103/physrevlett.80.65 1998

  36. [36]

    Shen, A new fast Chebyshev–Fourier algorithm for Poisson-type equations in polar geometries , Applied Numerical Mathematics, 33 (2000), pp

    J. Shen, A new fast Chebyshev–Fourier algorithm for Poisson-type equations in polar geometries , Applied Numerical Mathematics, 33 (2000), pp. 183–190, https://doi.org/10.1016/S0168-9274(99)00082-3

    work page doi:10.1016/s0168-9274(99)00082-3 2000

  37. [37]

    Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions , SIAM J

    J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions , SIAM J. Numer. Anal., 38 (2000), pp. 1113–1133, https://doi.org/10/dmwjs3

    work page 2000

  38. [38]

    J. Shen, T. Tang, and L.-L. Wang , Spectral Methods: Algorithms, Analysis and Applications , vol. 41, Springer Science & Business Media, 2011

    work page 2011

  39. [39]

    Shen and L.-L

    J. Shen and L.-L. Wang , Some recent advances on spectral methods for unbounded domains , Commun. Comput. Phys, 5 (2009), pp. 195–241

    work page 2009

  40. [40]

    Shen and Z.-Q

    J. Shen and Z.-Q. Wang, Error analysis of the strang time-splitting Laguerre–Hermite/Hermite collocation methods for the Gross–Pitaevskii equation , Foundations of Computational Mathematics, 13 (2013), 22 H. HU AND H. YU pp. 99–137, https://doi.org/10/gr7h2p

    work page 2013

  41. [41]

    Sheng, S

    C. Sheng, S. Ma, H. Li, L.-L. Wang, and L. Jia , Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schr¨ odinger operators, ESAIM: M2AN, 55 (2021), pp. 2141–2168, https://doi.org/10.1051/m2an/2021049

    work page doi:10.1051/m2an/2021049 2021

  42. [42]

    Stenger, Numerical Methods Based on Sinc and Analytic Functions , Springer New York, 1993

    F. Stenger, Numerical Methods Based on Sinc and Analytic Functions , Springer New York, 1993

    work page 1993

  43. [43]

    Sugihara, Optimality of the double exponential formula – functional analysis approach – , Numerische Mathematik, 75 (1997), pp

    M. Sugihara, Optimality of the double exponential formula – functional analysis approach – , Numerische Mathematik, 75 (1997), pp. 379–395

    work page 1997

  44. [44]

    Tang, The Hermite spectral method for Gaussian-type functions, SIAM Journal on Scientific Computing, 14 (1993), pp

    T. Tang, The Hermite spectral method for Gaussian-type functions, SIAM Journal on Scientific Computing, 14 (1993), pp. 594–606

    work page 1993

  45. [45]

    Tang and T

    T. Tang and T. Zhou , On discrete least-squares projection in unbounded domain with random evalua- tions and its application to parametric uncertainty quantification , SIAM J. Sci. Comput., 36 (2014), pp. A2272–A2295, https://doi.org/10.1137/140961894

    work page doi:10.1137/140961894 2014

  46. [46]

    L. N. Trefethen, Exactness of quadrature formulas , SIAM Review, 64 (2022), pp. 132–150

    work page 2022

  47. [47]

    Venturi, X

    D. Venturi, X. Wan, R. Mikulevicius, B. L. Rozovskii, and G. E. Karniadakis , Wick-Malliavin ap- proximation to nonlinear stochastic partial differential equations: analysis and simulations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), pp. 20130001– 20130001, https://doi.org/10.1098/rspa.2013.0001

    work page doi:10.1098/rspa.2013.0001 2013

  48. [48]

    Wan and G

    X. Wan and G. E. Karniadakis, A sharp error estimate for the fast Gauss transform , Journal of Compu- tational Physics, 219 (2006), pp. 7–12, https://doi.org/10.1016/j.jcp.2006.04.016

    work page doi:10.1016/j.jcp.2006.04.016 2006

  49. [49]

    Wan and H

    X. Wan and H. Yu, Numerical approximation of elliptic problems with log-normal random coefficients , Int. J. Uncertain. Quan., 9 (2019), https://doi.org/10.1615/Int.J.UncertaintyQuantification.2019029046

    work page doi:10.1615/int.j.uncertaintyquantification.2019029046 2019

  50. [50]

    Wang, Convergence analysis of Laguerre approximations for analytic functions, Math

    H. Wang, Convergence analysis of Laguerre approximations for analytic functions, Math. Comp., 93 (2024), pp. 2861–2884, https://doi.org/10.1090/mcom/3942

    work page doi:10.1090/mcom/3942 2024

  51. [51]

    Wang and L

    H. Wang and L. Zhang , Convergence analysis of Hermite approximations for analytic functions , 2023, https://doi.org/10.48550/arXiv.2312.07940

    work page doi:10.48550/arxiv.2312.07940 2023

  52. [52]

    M. Xia, S. Shao, and T. Chou , Efficient scaling and moving techniques for spectral methods in unbounded domains, SIAM J. Sci. Comput., 43 (2021), pp. A3244–A3268, https://doi.org/10/gsskw5

    work page 2021

  53. [53]

    Xiu and G

    D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos , Journal of Computational Physics, 187 (2003), pp. 137–167, https://doi.org/10.1016/S0021-9991(03) 00092-5

    work page doi:10.1016/s0021-9991(03 2003

  54. [54]

    Zhang, M

    G. Zhang, M. Gunzburger, and W. Zhao , A sparse-grid method for multi-dimensional backward sto- chastic differential equations, J. Comput. Math., 31 (2013), pp. 221–248, https://doi.org/10.4208/jcm. 1212-m4014

    work page doi:10.4208/jcm 2013

  55. [55]

    Zhang, L.-L

    X. Zhang, L.-L. Wang, and H. Jia , Error analysis of Fourier–Legendre and Fourier–Hermite spectral- Galerkin methods for the Vlasov–Poisson system , ESAIM: M2AN, 57 (2023), https://doi.org/10.1051/ m2an/2023091

    work page arXiv 2023