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arxiv: 1607.08538 · v2 · submitted 2016-07-28 · 🌌 astro-ph.GA

Recognition: 2 theorem links

Improving the full spectrum fitting method: accurate convolution with Gauss-Hermite functions

Michele Cappellari (University of Oxford)
Authors on Pith no claims yet

Pith reviewed 2026-05-09 17:50 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords full spectrum fittingGauss-Hermiteline-of-sight velocity distributionconvolutionFourier transformvelocity dispersiongalaxy kinematicspPXF
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The pith

Analytic Fourier transform of Gauss-Hermite kernels enables accurate convolution for low velocity dispersions in galaxy spectra

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

The paper improves the penalized pixel-fitting method used to extract stellar and gas kinematics from galaxy spectra via full-spectrum fitting. When velocity dispersion falls below the velocity sampling interval, which is set near the instrumental resolution, the standard practice of discretizing the convolution kernel produces large errors. The new approach replaces that discretization by directly evaluating the analytic Fourier transform of the Gauss-Hermite line-of-sight velocity distribution and applying the convolution theorem. The result is reliable kinematic measurements and redshifts even when dispersion is much smaller than the sampling, including for individual stars.

Core claim

The Gauss-Hermite parametrization of the line-of-sight velocity distribution possesses a closed-form Fourier transform that remains stable for any dispersion value. Substituting this transform into the convolution theorem performs the required smoothing of the template spectra without ever constructing an undersampled kernel on the observed velocity grid, removing the previous accuracy floor set by the ratio of dispersion to sampling interval.

What carries the argument

Analytic Fourier transform of the Gauss-Hermite series for the line-of-sight velocity distribution, substituted into the convolution theorem to replace discretized kernel evaluation.

If this is right

  • Accurate extraction of stellar and gas kinematics remains possible when dispersion is less than half the velocity sampling interval.
  • Galaxy redshifts can be measured reliably from spectra in which the dispersion lies well below the instrumental resolution.
  • Line-of-sight velocities of individual stars become measurable without the previous discretization bias.
  • Gaussian convolution routines used in other spectral-fitting packages can be replaced by the same analytic-transform approach.

Where Pith is reading between the lines

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

  • The same Fourier-domain strategy may apply to other parametric forms of the line-of-sight velocity distribution provided their transforms can be derived analytically.
  • Large spectroscopic surveys could reduce template oversampling requirements, lowering memory and CPU costs while preserving accuracy at low dispersion.
  • The method opens a path to hybrid parametric-nonparametric fitting in which the Fourier representation handles the convolution step uniformly.

Load-bearing premise

The analytic Fourier transform of the Gauss-Hermite kernel can be computed numerically with sufficient accuracy and stability for all relevant dispersion values, and its insertion into the convolution theorem does not introduce new fitting artifacts.

What would settle it

Generate synthetic spectra with known input velocities and dispersions much smaller than the pixel sampling, then check whether the recovered parameters agree with the inputs to within the formal uncertainties.

read the original abstract

I start by providing an updated summary of the penalized pixel-fitting (pPXF) method, which is used to extract the stellar and gas kinematics, as well as the stellar population of galaxies, via full spectrum fitting. I then focus on the problem of extracting the kinematic when the velocity dispersion $\sigma$ is smaller than the velocity sampling $\Delta V$, which is generally, by design, close to the instrumental dispersion $\sigma_{\rm inst}$. The standard approach consists of convolving templates with a discretized kernel, while fitting for its parameters. This is obviously very inaccurate when $\sigma<\Delta V/2$, due to undersampling. Oversampling can prevent this, but it has drawbacks. Here I present a more accurate and efficient alternative. It avoids the evaluation of the under-sampled kernel, and instead directly computes its well-sampled analytic Fourier transform, for use with the convolution theorem. A simple analytic transform exists when the kernel is described by the popular Gauss-Hermite parametrization (which includes the Gaussian as special case) for the line-of-sight velocity distribution. I describe how this idea was implemented in a significant upgrade to the publicly available pPXF software. The key advantage of the new approach is that it provides accurate velocities regardless of $\sigma$. This is important e.g. for spectroscopic surveys targeting galaxies with $\sigma\ll\sigma_{\rm inst}$, for galaxy redshift determinations, or for measuring line-of-sight velocities of individual stars. The proposed method could also be used to fix Gaussian convolution algorithms used in today's popular software packages.

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

0 major / 2 minor

Summary. The manuscript updates the description of the penalized pixel-fitting (pPXF) method for extracting stellar/gas kinematics and stellar populations via full-spectrum fitting. It identifies the inaccuracy of discretized pixel-space convolution with a Gauss-Hermite LOSVD kernel when velocity dispersion σ is smaller than the sampling interval ΔV (typically near the instrumental dispersion). The proposed solution replaces this with direct evaluation of the analytic Fourier transform of the Gauss-Hermite kernel (including the Gaussian limit) followed by application of the convolution theorem. The approach is implemented as a significant upgrade to the public pPXF software, with the central claim that it yields accurate velocities independent of σ.

Significance. If the numerical implementation holds, the result would be significant for kinematic analyses in spectroscopic surveys, especially for systems with σ ≪ σ_inst, precise galaxy redshifts, and line-of-sight velocities of individual stars. The method exploits the known closed-form Fourier transform property of Hermite-Gaussian functions (mapping to the same functional class up to phase and scaling), providing an efficient, non-approximate alternative to oversampling without introducing new mathematical approximations beyond the discrete FFT already used in spectral processing. The public code release supports reproducibility.

minor comments (2)
  1. [Method description] The abstract states that the method 'avoids the evaluation of the under-sampled kernel'; a brief explicit statement in the main text (near the description of the convolution theorem application) confirming that no additional truncation or windowing is introduced would strengthen clarity.
  2. [Figures] Figure captions and axis labels should explicitly note the range of σ/ΔV values tested to allow readers to assess coverage of the low-σ regime highlighted in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of our contributions, and the recommendation to accept. We are pleased that the significance for kinematic analyses in spectroscopic surveys, especially when σ ≪ σ_inst, was recognized.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central contribution is the replacement of undersampled pixel-space convolution with the analytic Fourier transform of the Gauss-Hermite LOSVD (including the Gaussian limit) applied via the convolution theorem. This rests on the standard mathematical fact that the Fourier transform of Hermite-Gaussian functions remains within the same functional class (up to scaling and phase), which is an external property of the functions and not derived from or equivalent to any fitted parameters, self-definitions, or prior claims within the paper. The summary of the existing pPXF method provides context but does not bear the load of the new accuracy claim for σ ≪ ΔV; that claim follows directly from the discrete FFT implementation of a known transform. No steps reduce by construction to inputs, no uniqueness theorems are imported from self-citations, and no ansatz is smuggled. The derivation is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of Fourier transforms and the established use of Gauss-Hermite series for line-of-sight velocity distributions in stellar kinematics.

axioms (2)
  • standard math The convolution theorem applies to the Fourier transform of the Gauss-Hermite kernel.
    Invoked to justify performing convolution via multiplication in Fourier space.
  • domain assumption Gauss-Hermite parametrization is a valid description of the line-of-sight velocity distribution.
    Standard assumption in the pPXF method for extracting galaxy kinematics.

pith-pipeline@v0.9.0 · 5583 in / 1315 out tokens · 31436 ms · 2026-05-09T17:50:11.440405+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

97 extracted references · 97 canonical work pages · cited by 17 Pith papers · 1 internal anchor

  1. [1]

    A., 1964, Handbook of Mathematical Functions (Reprinted 1972)

    Abramowitz M., Stegun I. A., 1964, Handbook of Mathematical Functions (Reprinted 1972). National Bureau of Standards, Washington, https://books.google.com/books?id=MtU8uP7XMvoC

    work page 1964

  2. [2]

    Alatalo K., et al., 2011, @doi [ ] 10.1088/0004-637X/735/2/88 , http://adsabs.harvard.edu/abs/2011ApJ...735...88A 735, 88

    work page doi:10.1088/0004-637x/735/2/88 2011

  3. [3]

    Astropy Collaboration 2013, @doi [ ] 10.1051/0004-6361/201322068 , http://adsabs.harvard.edu/abs/2013A

    work page doi:10.1051/0004-6361/201322068 2013

  4. [4]

    K., et al., 2015, @doi [ ] 10.1051/0004-6361/201424935 , http://adsabs.harvard.edu/abs/2015A

    Barrera-Ballesteros J. K., et al., 2015, @doi [ ] 10.1051/0004-6361/201424935 , http://adsabs.harvard.edu/abs/2015A

    work page doi:10.1051/0004-6361/201424935 2015

  5. [5]

    P., Gerhard O

    Bender R., Saglia R. P., Gerhard O. E., 1994, , http://adsabs.harvard.edu/abs/1994MNRAS.269..785B 269, 785

    work page 1994

  6. [6]

    R., Shen S

    Berger H., Simental E., 1999, in Descour M. R., Shen S. S., eds, \ Vol. 3753, Imaging Spectrometry V. pp 124--132, @doi 10.1117/12.366275

    work page doi:10.1117/12.366275 1999

  7. [7]

    and Verheijen, Marc A

    Bershady M. A., Verheijen M. A. W., Swaters R. A., Andersen D. R., Westfall K. B., Martinsson T., 2010, @doi [ ] 10.1088/0004-637X/716/1/198 , http://adsabs.harvard.edu/abs/2010ApJ...716..198B 716, 198

    work page doi:10.1088/0004-637x/716/1/198 2010

  8. [8]

    A., et al., 2013, @doi [ ] 10.1088/0004-6256/145/5/138 , http://adsabs.harvard.edu/abs/2013AJ....145..138B 145, 138

    Blanc G. A., et al., 2013, @doi [ ] 10.1088/0004-6256/145/5/138 , http://adsabs.harvard.edu/abs/2013AJ....145..138B 145, 138

    work page doi:10.1088/0004-6256/145/5/138 2013

  9. [9]

    A., Coleman , T

    Branch M. A., Coleman T. F., Li Y., 1999, @doi [SIAM Journal on Scientific Computing] 10.1137/S1064827595289108 , 21, 1

    work page doi:10.1137/s1064827595289108 1999

  10. [10]

    O., 1974, The fast Fourier transform

    Brigham E. O., 1974, The fast Fourier transform. Prentice-Hall Inc., Englewood Cliffs, NJ

    work page 1974

  11. [11]

    Bruzual G., Charlot S., 2003, @doi [ ] 10.1046/j.1365-8711.2003.06897.x , http://adsabs.harvard.edu/abs/2003MNRAS.344.1000B 344, 1000

    work page doi:10.1046/j.1365-8711.2003.06897.x 2003

  12. [12]

    J., Owers, M

    Bryant J. J., et al., 2015, @doi [ ] 10.1093/mnras/stu2635 , http://adsabs.harvard.edu/abs/2015MNRAS.447.2857B 447, 2857

    work page doi:10.1093/mnras/stu2635 2015

  13. [13]

    Bundy K., et al., 2015, @doi [ ] 10.1088/0004-637X/798/1/7 , http://adsabs.harvard.edu/abs/2015ApJ...798....7B 798, 7

    work page Pith review doi:10.1088/0004-637x/798/1/7 2015

  14. [14]

    C., et al

    Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, @doi [ ] 10.1086/308692 , http://adsabs.harvard.edu/abs/2000ApJ...533..682C 533, 682

    work page Pith review doi:10.1086/308692 2000

  15. [15]

    Cappellari M., 2016, @doi [ ] 10.1146/annurev-astro-082214-122432 , http://adsabs.harvard.edu/abs/2016ARA

    work page doi:10.1146/annurev-astro-082214-122432 2016

  16. [16]

    Cappellari M., Emsellem E., 2004, @doi [ ] 10.1086/381875 , http://adsabs.harvard.edu/abs/2004PASP..116..138C 116, 138

    work page doi:10.1086/381875 2004

  17. [17]

    Cappellari M., et al., 2009, @doi [ ] 10.1088/0004-637X/704/1/L34 , http://adsabs.harvard.edu/abs/2009ApJ...704L..34C 704, L34

    work page doi:10.1088/0004-637x/704/1/l34 2009

  18. [18]

    Cappellari M., et al., 2011, @doi [ ] 10.1111/j.1365-2966.2010.18174.x , http://adsabs.harvard.edu/abs/2011MNRAS.413..813C 413, 813

    work page doi:10.1111/j.1365-2966.2010.18174.x 2011

  19. [19]

    Cappellari M., et al., 2012, @doi [ ] 10.1038/nature10972 , http://adsabs.harvard.edu/abs/2012Natur.484..485C 484, 485

    work page doi:10.1038/nature10972 2012

  20. [20]

    Cappellari M., et al., 2013a, @doi [ ] 10.1093/mnras/stt562 , http://adsabs.harvard.edu/abs/2013MNRAS.432.1709C 432, 1709

    work page doi:10.1093/mnras/stt562

  21. [21]

    Cappellari M., et al., 2013b, @doi [ ] 10.1093/mnras/stt644 , http://adsabs.harvard.edu/abs/2013MNRAS.432.1862C 432, 1862

    work page doi:10.1093/mnras/stt644

  22. [22]

    Cappellari M., et al., 2015, @doi [ ] 10.1088/2041-8205/804/1/L21 , http://adsabs.harvard.edu/abs/2015ApJ...804L..21C 804, L21

    work page doi:10.1088/2041-8205/804/1/l21 2015

  23. [23]

    A., Clayton, G

    Cardelli J. A., Clayton G. C., Mathis J. S., 1989, @doi [ ] 10.1086/167900 , http://adsabs.harvard.edu/abs/1989ApJ...345..245C 345, 245

    work page doi:10.1086/167900 1989

  24. [24]

    2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x

    Cenarro A. J., Cardiel N., Gorgas J., Peletier R. F., Vazdekis A., Prada F., 2001, @doi [ ] 10.1046/j.1365-8711.2001.04688.x , http://adsabs.harvard.edu/abs/2001MNRAS.326..959C 326, 959

    work page doi:10.1046/j.1365-8711.2001.04688.x 2001

  25. [25]

    Cheung E., et al., 2016, @doi [ ] 10.1038/nature18006 , http://adsabs.harvard.edu/abs/2016Natur.533..504C 533, 504

    work page doi:10.1038/nature18006 2016

  26. [26]

    2005 , month = sep, journal =

    Cid Fernandes R., Mateus A., Sodr \'e L., Stasi \'n ska G., Gomes J. M., 2005, @doi [ ] 10.1111/j.1365-2966.2005.08752.x , http://adsabs.harvard.edu/abs/2005MNRAS.358..363C 358, 363

    work page doi:10.1111/j.1365-2966.2005.08752.x 2005

  27. [27]

    P., Castilho B

    Coelho P., Barbuy B., Mel \'e ndez J., Schiavon R. P., Castilho B. V., 2005, @doi [ ] 10.1051/0004-6361:20053511 , http://adsabs.harvard.edu/abs/2005A

    work page doi:10.1051/0004-6361:20053511 2005

  28. [28]

    Conroy C., 2013, @doi [ ] 10.1146/annurev-astro-082812-141017 , http://adsabs.harvard.edu/abs/2013ARA

    work page Pith review doi:10.1146/annurev-astro-082812-141017 2013

  29. [29]

    Conroy C., van Dokkum P., 2012, @doi [ ] 10.1088/0004-637X/747/1/69 , http://adsabs.harvard.edu/abs/2012ApJ...747...69C 747, 69

    work page doi:10.1088/0004-637x/747/1/69 2012

  30. [30]

    Cooley and John W

    Cooley J. W., Tukey J. W., 1965, @doi [Mathematics of computation] 10.2307/2003354 , 19, 297

    work page doi:10.2307/2003354 1965

  31. [31]

    W., & Ebeling, H

    Davis T. A., et al., 2011, @doi [ ] 10.1111/j.1365-2966.2011.19355.x , http://adsabs.harvard.edu/abs/2011MNRAS.417..882D 417, 882

    work page doi:10.1111/j.1365-2966.2011.19355.x 2011

  32. [32]

    Emsellem E., et al., 2004, @doi [ ] 10.1111/j.1365-2966.2004.07948.x , http://adsabs.harvard.edu/abs/2004MNRAS.352..721E 352, 721

    work page doi:10.1111/j.1365-2966.2004.07948.x 2004

  33. [33]

    Emsellem E., et al., 2011, @doi [ ] 10.1111/j.1365-2966.2011.18496.x , http://adsabs.harvard.edu/abs/2011MNRAS.414..888E 414, 888

    work page doi:10.1111/j.1365-2966.2011.18496.x 2011

  34. [34]

    B., Stern H

    Gelman A., Carlin J. B., Stern H. S., Dunson D. B., Vehtari A., Rubin D. B., 2013, Bayesian data analysis, 3rd edition. Chapman & Hall/CRC, Boca Raton, https://books.google.com/books?id=ZXL6AQAAQBAJ

    work page 2013

  35. [35]

    E., 1993, , http://adsabs.harvard.edu/abs/1993MNRAS.265..213G 265, 213

    Gerhard O. E., 1993, , http://adsabs.harvard.edu/abs/1993MNRAS.265..213G 265, 213

    work page 1993

  36. [36]

    Getreuer P., 2013, @doi [Image Processing On Line] 10.5201/ipol.2013.87 , 3, 286

    work page doi:10.5201/ipol.2013.87 2013

  37. [37]

    S., Ryzhik I

    Gradshteyn I. S., Ryzhik I. M., 1980, Table of integrals, series, and products (8th ed.\ 2014). Academic Press, New York, @doi 10.1016/B978-0-12-384933-5.00008-4

    work page doi:10.1016/b978-0-12-384933-5.00008-4 1980

  38. [38]

    2008, A&A, 486, 951, doi: 10.1051/0004-6361:200809724

    Gustafsson B., Edvardsson B., Eriksson K., J rgensen U. G., Nordlund ., Plez B., 2008, @doi [ ] 10.1051/0004-6361:200809724 , http://adsabs.harvard.edu/abs/2008A

    work page doi:10.1051/0004-6361:200809724 2008

  39. [39]

    C., 1998, Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion

    Hansen P. C., 1998, Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. Mathematical Modeling and Computation Vol. 4, Siam, Philadelphia, @doi 10.1137/1.9780898719697

    work page doi:10.1137/1.9780898719697 1998

  40. [40]

    C., Filippenko , A

    Ho L. C., Filippenko A. V., Sargent W. L. W., 1997, @doi [ ] 10.1086/313041 , http://adsabs.harvard.edu/abs/1997ApJS..112..315H 112, 315

    work page doi:10.1086/313041 1997

  41. [41]

    Ho I.-T., et al., 2016, @doi [ ] 10.1093/mnras/stw017 , http://adsabs.harvard.edu/abs/2016MNRAS.457.1257H 457, 1257

    work page doi:10.1093/mnras/stw017 2016

  42. [42]

    Distance measures in cosmology

    Hogg D. W., 1999, preprint, http://adsabs.harvard.edu/abs/1999astro.ph..5116H ( @eprint astro-ph/9905116 )

    work page internal anchor Pith review arXiv 1999

  43. [43]

    Computing in Science & Engineering9(3), 90–95 (2007) https://doi.org/10.1109/MCSE.2007.55

    Hunter J. D., 2007, @doi [Computing In Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    work page doi:10.1109/mcse.2007.55 2007

  44. [44]

    J., Merrifield M

    Johnston E. J., Merrifield M. R., Arag \'o n-Salamanca A., Cappellari M., 2013, @doi [ ] 10.1093/mnras/sts121 , http://adsabs.harvard.edu/abs/2013MNRAS.428.1296J 428, 1296

    work page doi:10.1093/mnras/sts121 2013

  45. [45]

    Jones E., Oliphant T., Peterson P., et al., 2001, SciPy : Open source scientific tools for Python , http://www.scipy.org/

    work page 2001

  46. [46]

    D., Illingworth G

    Kelson D. D., Illingworth G. D., van Dokkum P. G., Franx M., 2000, @doi [ ] 10.1086/308445 , http://adsabs.harvard.edu/abs/2000ApJ...531..159K 531, 159

    work page doi:10.1086/308445 2000

  47. [47]

    2005 , month = sep, journal =

    Krajnovi \'c D., Cappellari M., de Zeeuw P. T., Copin Y., 2006, @doi [ ] 10.1111/j.1365-2966.2005.09902.x , http://adsabs.harvard.edu/abs/2006MNRAS.366..787K 366, 787

    work page doi:10.1111/j.1365-2966.2005.09902.x 2006

  48. [48]

    MNRAS403(2), 625–645 (2010) https://doi.org/10.1111/j.1365-2966.2009

    Krajnovi \'c D., McDermid R. M., Cappellari M., Davies R. L., 2009, @doi [ ] 10.1111/j.1365-2966.2009.15415.x , http://adsabs.harvard.edu/abs/2009MNRAS.399.1839K 399, 1839

    work page doi:10.1111/j.1365-2966.2009.15415.x 2009

  49. [49]

    L., 2005, Memorie della Societa Astronomica Italiana Supplementi, http://adsabs.harvard.edu/abs/2005MSAIS...8..189K 8, 189

    Kurucz R. L., 2005, Memorie della Societa Astronomica Italiana Supplementi, http://adsabs.harvard.edu/abs/2005MSAIS...8..189K 8, 189

    work page 2005

  50. [50]

    L., Hanson R

    Lawson C. L., Hanson R. J., 1974, Solving least squares problems (SIAM 1995 edition). Classics in applied mathematics Vol. 15, Prentice-Hall Inc., Englewood Cliffs, NJ, @doi 10.1137/1.9781611971217

    work page doi:10.1137/1.9781611971217 1974

  51. [51]

    Maraston C., Str \"o mb \"a ck G., 2011, @doi [ ] 10.1111/j.1365-2966.2011.19738.x , http://adsabs.harvard.edu/abs/2011MNRAS.418.2785M 418, 2785

    work page doi:10.1111/j.1365-2966.2011.19738.x 2011

  52. [52]

    astro-ph.IM

    Markwardt C. B., 2009, in D. A. Bohlender D. Durand . P. D., ed., Astronomical Society of the Pacific Conference Series Vol. 411, Astronomical Data Analysis Software and Systems XVIII. p. 251 ( @eprint arXiv 0902.2850 )

    work page arXiv 2009

  53. [53]

    M., et al., 2015, @doi [ ] 10.1093/mnras/stv105 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.448.3484M 448, 3484

    McDermid R. M., et al., 2015, @doi [ ] 10.1093/mnras/stv105 , http://adsabs.harvard.edu/abs/2015MNRAS.448.3484M 448, 3484

    work page doi:10.1093/mnras/stv105 2015

  54. [54]

    J., 2016, preprint, http://adsabs.harvard.edu/abs/2016arXiv161004516M ( @eprint arXiv 1610.04516 )

    Mitzkus M., Cappellari M., Walcher C. J., 2016, preprint, http://adsabs.harvard.edu/abs/2016arXiv161004516M ( @eprint arXiv 1610.04516 )

    work page arXiv 2016

  55. [55]

    Argonne National Laboratory Argonne, IL, http://cds.cern.ch/record/126569

    Mor \'e J., Garbow B., Hillstrom K., 1980, User guide for MINPACK-1. Argonne National Laboratory Argonne, IL, http://cds.cern.ch/record/126569

    work page 1980

  56. [56]

    Morelli L., Calvi V., Masetti N., Parisi P., Landi R., Maiorano E., Minniti D., Galaz G., 2013, @doi [ ] 10.1051/0004-6361/201321733 , http://adsabs.harvard.edu/abs/2013A

    work page doi:10.1051/0004-6361/201321733 2013

  57. [57]

    M., Pizzella A., Dalla Bont \`a E., Coccato L., M \'e ndez-Abreu J., 2015, @doi [ ] 10.1093/mnras/stv1357 , http://adsabs.harvard.edu/abs/2015MNRAS.452.1128M 452, 1128

    Morelli L., Corsini E. M., Pizzella A., Dalla Bont \`a E., Coccato L., M \'e ndez-Abreu J., 2015, @doi [ ] 10.1093/mnras/stv1357 , http://adsabs.harvard.edu/abs/2015MNRAS.452.1128M 452, 1128

    work page doi:10.1093/mnras/stv1357 2015

  58. [58]

    Munari U., Sordo R., Castelli F., Zwitter T., 2005, , 442, 1127

    work page 2005

  59. [59]

    Naab T., et al., 2014, @doi [ ] 10.1093/mnras/stt1919 , http://adsabs.harvard.edu/abs/2014MNRAS.444.3357N 444, 3357

    work page doi:10.1093/mnras/stt1919 2014

  60. [60]

    Springer Series in Operations Research and Financial Engineering, Springer, New York, https://books.google.com/books?id=VbHYoSyelFcC

    Nocedal J., Wright S., 2006, Numerical Optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, https://books.google.com/books?id=VbHYoSyelFcC

    work page 2006

  61. [61]

    Ocvirk P., Pichon C., Lan c on A., Thi \'e baut E., 2006, @doi [ ] 10.1111/j.1365-2966.2005.09182.x , http://adsabs.harvard.edu/abs/2006MNRAS.365...46O 365, 46

    work page doi:10.1111/j.1365-2966.2005.09182.x 2006

  62. [62]

    K., 2011, @doi [ ] 10.1088/0067-0049/195/2/13 , http://adsabs.harvard.edu/abs/2011ApJS..195...13O 195, 13

    Oh K., Sarzi M., Schawinski K., Yi S. K., 2011, @doi [ ] 10.1088/0067-0049/195/2/13 , http://adsabs.harvard.edu/abs/2011ApJS..195...13O 195, 13

    work page doi:10.1088/0067-0049/195/2/13 2011

  63. [63]

    E., 2007, @doi [Computing in Science Engineering] 10.1109/MCSE.2007.58 , 9, 10

    Oliphant T. E., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.58 , 9, 10

    work page doi:10.1109/mcse.2007.58 2007

  64. [64]

    Onodera M., et al., 2012, @doi [ ] 10.1088/0004-637X/755/1/26 , http://adsabs.harvard.edu/abs/2012ApJ...755...26O 755, 26

    work page doi:10.1088/0004-637x/755/1/26 2012

  65. [65]

    H., Teukolsky S

    Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., 2007, Numerical recipes: The art of scientific computing, 3rd edn. Cambridge Univ. Press, Cambridge, https://books.google.com/books?id=1aAOdzK3FegC

    work page 2007

  66. [66]

    Prugniel P., Soubiran C., 2001, @doi [ ] 10.1051/0004-6361:20010163 , http://adsabs.harvard.edu/abs/2001A

    work page doi:10.1051/0004-6361:20010163 2001

  67. [67]

    Rix H.-W., White S. D. M., 1992, , http://ads.ari.uni-heidelberg.de/abs/1992MNRAS.254..389R 254, 389

    work page 1992

  68. [68]

    SDSS Collaboration 2016, preprint, http://adsabs.harvard.edu/abs/2016arXiv160802013S ( @eprint arXiv 1608.02013 )

    work page arXiv 2016

  69. [69]

    S \'a nchez-Bl \'a zquez P., et al., 2006, @doi [ ] 10.1111/j.1365-2966.2006.10699.x , http://adsabs.harvard.edu/abs/2006MNRAS.371..703S 371, 703

    work page doi:10.1111/j.1365-2966.2006.10699.x 2006

  70. [70]

    F., et al., 2012, @doi [ ] 10.1051/0004-6361/201117353 , http://adsabs.harvard.edu/abs/2012A

    S \'a nchez S. F., et al., 2012, @doi [ ] 10.1051/0004-6361/201117353 , http://adsabs.harvard.edu/abs/2012A

    work page doi:10.1051/0004-6361/201117353 2012

  71. [71]

    Sargent W. L. W., Schechter P. L., Boksenberg A., Shortridge K., 1977, @doi [ ] 10.1086/155052 , http://adsabs.harvard.edu/abs/1977ApJ...212..326S 212, 326

    work page doi:10.1086/155052 1977

  72. [72]

    2005 , month = sep, journal =

    Sarzi M., Falc \'o n-Barroso J., Davies R. L., et al. 2006, @doi [ ] 10.1111/j.1365-2966.2005.09839.x , http://adsabs.harvard.edu/abs/2006MNRAS.366.1151S 366, 1151

    work page doi:10.1111/j.1365-2966.2005.09839.x 2006

  73. [73]

    L., Gunn J

    Schechter P. L., Gunn J. E., 1979, @doi [ ] 10.1086/156978 , http://adsabs.harvard.edu/abs/1979ApJ...229..472S 229, 472

    work page doi:10.1086/156978 1979

  74. [74]

    I., 1968, Quantum Mechanics 3rd ed

    Schiff L. I., 1968, Quantum Mechanics 3rd ed.. McGraw-Hill, New York

    work page 1968

  75. [75]

    Scott N., et al., 2015, @doi [ ] 10.1093/mnras/stv1127 , http://adsabs.harvard.edu/abs/2015MNRAS.451.2723S 451, 2723

    work page doi:10.1093/mnras/stv1127 2015

  76. [76]

    C., van den Bosch, R., Mieske, S., et al

    Seth A. C., et al., 2014, @doi [ ] 10.1038/nature13762 , http://adsabs.harvard.edu/abs/2014Natur.513..398S 513, 398

    work page doi:10.1038/nature13762 2014

  77. [77]

    Shetty S., Cappellari M., 2015, @doi [ ] 10.1093/mnras/stv1948 , http://adsabs.harvard.edu/abs/2015MNRAS.454.1332S 454, 1332

    work page doi:10.1093/mnras/stv1948 2015

  78. [78]

    J., & Hummer, D

    Storey P. J., Hummer D. G., 1995, @doi [ ] 10.1093/mnras/272.1.41 , http://adsabs.harvard.edu/abs/1995MNRAS.272...41S 272, 41

    work page doi:10.1093/mnras/272.1.41 1995

  79. [79]

    N., 1982, in Rodrigue G., ed., Parallel Computations

    Swarztrauber P. N., 1982, in Rodrigue G., ed., Parallel Computations. Academic Press, New York, pp 51--83, @doi 10.1016/B978-0-12-592101-5.50007-5

    work page doi:10.1016/b978-0-12-592101-5.50007-5 1982

  80. [80]

    P., Cappellari M., Johnston E., 2016, submitted to

    Tabor M., Merrifield M., Arag\'on-Salamanca A., Bamford S. P., Cappellari M., Johnston E., 2016, submitted to

    work page 2016

Showing first 80 references.