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Richard S. Laugesen - Research

Research Areas

1. Eigenvalues of differential operators (esp. Laplacian and bi-Laplacian)
Eigenvalues represent stability indices, frequencies of vibration, rates of decay to equilibrium, and quantum energy levels. I like to prove sharp inequalities.

Click for open problems from workshops at the: American Institute of Mathematics, the Oberwolfach Research Institute (pages 45-71), and the Queen Dido Conference.
Here is an overview of 2000 years of work on isoperimetric problems in mathematical physics, and a more technical survey of properties of eigenvalues and eigenfunctions of the Laplacian, by Grebenkov and Nguyen.

2. Wavelet and Gabor systems in applied harmonic analysis
How "bad" or "good" can a system of functions be, if the system is generated by simple geometric operations (translation, dilation, modulation) and is required to be capable of reproducing every possible signal in the standard function spaces of analysis (Lp, Sobolev, Hardy)?

Publications

  1. The Robin Laplacian --- spectral conjectures, rectangular theorems
    Submitted. 44 pages.
    preprint (correction: in Conjecture D, the convex domain should have smooth boundary)

  2. BIG career developments for mathematics graduate students
    (With R. Levy and F. Santosa.)
    Notices of the American Mathematical Society, 66 (2019) no. 4, 523-524.
    reprint

  3. Symmetrization in Analysis
    (By Albert Baernstein II, with David Drasin and Richard Laugesen.)
    Cambridge University Press, 2019. 474 pages.
    purchase

  4. From Steklov to Neumann and beyond, via Robin: the Szegő way
    (With P. Freitas.)
    Canadian Journal of Mathematics, to appear. 20 pages.
    preprint (correction: in Conjecture 1, the convex domain should have smooth boundary)

  5. From Neumann to Steklov and beyond, via Robin: the Weinberger way
    (With P. Freitas.)
    Submitted. 23 pages.
    preprint

  6. BIG Jobs Guide: Business, Industry, and Government Careers for Mathematical Scientists, Statisticians, and Operations Researchers
    (By Rachel Levy, Richard Laugesen, and Fadil Santosa.)
    SIAM book series, 2018. 141 pages.
    purchase

  7. Spectral theory of partial differential equations
    In: Spectral Theory and Applications, Contemporary Mathematics, vol. 720, American Mathematical Society, Providence, RI, 2018, pp. 23-55. (Proceedings of the 2016 CRM Summer School on Spectral Theory and Applications.)

  8. Pólya's conjecture fails for the fractional Laplacian
    (With M. Kwaśnicki and B. A. Siudeja.)
    Journal of Spectral Theory, 9 (2019) no. 1, 127–135.
    reprint

  9. Shifted lattices and asymptotically optimal ellipses
    (With S. Liu.)
    The Journal of Analysis, 26 (2018) no. 1, 71-102.
    reprint

  10. Optimal stretching for lattice points and eigenvalues
    (With S. Liu.)
    Arkiv för Matematik, 56 (2018) no. 1, 111-145.
    reprint

  11. Optimal stretching for lattice points under convex curves
    (With S. Ariturk.)
    Portugaliae Mathematica (N.S.), 74 (2017) no. 2, 91-114.
    reprint

  12. Triangles and Other Special Domains
    (With B. A. Siudeja.)
    Chapter 6 (pp. 149-200) in the book Shape Optimization and Spectral Theory, edited by Antoine Henrot, De Gruyter Open, 2017.
    open access

  13. Math PhD careers: new opportunities emerging amidst crisis
    (With Y. Baryshnikov and R. DeVille.)
    Notices of the American Mathematical Society, 64 (2017) no. 3, 260-264.
    reprint

  14. Preparing graduates for careers in the mathematical sciences
    (With S. Minkoff, W. Menasco, F. Santosa, S. Pankavich.)
    SIAM News, 49 (2016).
    reprint

  15. Torsion and ground state maxima: close but not the same
    (With B. A. Benson, M. Minion and B. A. Siudeja.)
    Bulletin of the Irish Mathematical Society, 78 (2016), 81-88.
    reprint

  16. Steklov eigenvalues and quasiconformal maps of simply connected planar domains
    (With A. Girouard and B. A. Siudeja.)
    Archive for Rational Mechanics and Analysis, 219 (2016), 903-936.
    reprint

  17. Multivariable feedback particle filter
    (With T. Yang, P. G. Mehta and S. Meyn.)
    Automatica, 71 (2016), 10-23.
    reprint

  18. Poisson's equation in nonlinear filtering
    (With P. G. Mehta, S. Meyn, and M. Raginsky.)
    SIAM Journal on Control and Optimization, 53 (2015), 501-525.
    reprint

  19. Magnetic spectral bounds on starlike plane domains
    (With B. A. Siudeja.)
    ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 670-689.
    reprint

  20. Sharp spectral bounds on starlike domains
    (With B. A. Siudeja.)
    Journal of Spectral Theory, 4 (2014), 309-347.
    reprint

  21. Explicit interpolation bounds between Hardy space and L2
    (With H.-Q. Bui.)
    Journal of the Australian Mathematical Society, 95 (2013), 158-168.
    reprint

  22. Wavelet frame bijectivity on Lebesgue and Hardy spaces
    (With H.-Q. Bui.)
    Journal of Fourier Analysis and Applications 19 (2013), 376-409.
    reprint

  23. Multivariable feedback particle filter
    (With T. Yang, P. G. Mehta and S. Meyn.)
    2012 IEEE 51st Annual Conference on Decision and Control (CDC), (Dec. 2012), 4063-4070.
    reprint

  24. Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian
    In: Proceedings of the AMSI International Conference on Harmonic Analysis and Applications (Macquarie University, February 2011), pp. 63-82. Edited by X. Duong, J. Hogan, C. Meaney, A. Sikora.
    reprint

  25. Uniqueness for the continuous wavelet transform
    (With H.-Q. Bui.)
    Far East Journal of Applied Mathematics 65 (2012), 1-11.
    reprint

  26. Sums of magnetic eigenvalues are maximal on rotationally symmetric domains
    (With J. Liang and A. Roy.)
    Annales Henri Poincare 13 (2012), 731-750.
    reprint

  27. Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals
    (With Z. C. Pan and S. S. Son.)
    Mathematical Inequalities and Applications 15 (2012), 381-394.
    reprint

  28. Rebuttal of Donnelly's paper on the spectral gap
    (With M. S. Ashbaugh and A. Henrot.)
    Mathematische Zeitschrift 269 (2011), 5-7.
    reprint

  29. Dirichlet eigenvalue sums on triangles are minimal for equilaterals
    (With B. A. Siudeja.)
    Communications in Analysis and Geometry 19 (2011), 855--885.
    reprint

    AMS Fellows - a modest proposal
    AMS Notices, Letter to the Editor, December 2011.
    reprint
    Readers might enjoy the original "A Modest Proposal" by Jonathan Swift (1729).

  30. Sums of Laplace eigenvalues - rotations and tight frames in higher dimensions
    (With B. A. Siudeja.)
    Journal of Mathematical Physics 52 (2011), 093703. 13 pages.
    reprint

  31. Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane
    (With B. A. Siudeja.)
    Journal of Functional Analysis 260 (2011), 1795-1823.
    reprint

  32. Approximately dual frames in Hilbert spaces and applications to Gabor frames
    (With O. Christensen.)
    Sampling Theory in Signal and Image Processing 9 (2011), 77-90.
    preprint

  33. Wavelets in Littlewood-Paley space, and Mexican hat completeness
    (With H.-Q. Bui.)
    Applied and Computational Harmonic Analysis 30 (2011), 204-213.
    reprint (and typos)
    For related estimates, see our technical report A note on the solution of the Mexican hat problem.

  34. Frequency-scale frames and the solution of the Mexican hat problem
    (With H.-Q. Bui.)
    Constructive Approximation 33 (2011), 163–189.
    reprint

  35. Moment inequalities for equilibrium measures in the plane
    (With A. Baernstein II and I. E. Pritsker.)
    Pure and Applied Mathematics Quarterly 7 (2011), 51-86.
    reprint

  36. Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality
    (With B. A. Siudeja.)
    Journal of Differential Equations 249 (2010), 118-135.
    reprint

  37. Maximizing Neumann fundamental tones of triangles
    (With B. A. Siudeja.)
    22 pages. Journal of Mathematical Physics, 50:112903, 2009.
    reprint

  38. Gabor dual spline windows
    Applied and Computational Harmonic Analysis, 27:180-194, 2009.
    reprint, or an earlier preprint with more details

  39. A computable Fourier condition generating alias-free sampling lattices
    (With Y. M. Lu and M. N. Do.)
    IEEE Transactions on Signal Processing, 57:1768-1782, 2009.
    reprint

  40. A note on constructing affine systems for L2
    (With H.-Q. Bui and N. Kaiblinger.)
    Applied and Computational Harmonic Analysis, 25:400-406, 2008.
    reprint

  41. Affine synthesis onto Lp when 0 < p <= 1
    Journal of Fourier Analysis and Applications, 14:235-266, 2008.
    reprint

  42. Affine synthesis onto Lebesgue and Hardy spaces
    (With H.-Q. Bui.)
    Indiana University Mathematics Journal, 57:2203-2233, 2008.
    reprint; an earlier version that treats Sobolev spaces too is here

  43. Sobolev spaces and approximation by affine spanning systems
    (With H.-Q. Bui.)
    Mathematische Annalen, 341:347-389, 2008.
    reprint

  44. Approximation and spanning in the Hardy space, by affine systems
    (With H.-Q. Bui.)
    Constructive Approximation, 28:149-172, 2008.
    reprint

  45. On convex surfaces with minimal moment of inertia
    (With P. Freitas and G. F. Liddell.)
    Journal of Mathematical Physics, 48, 122902, 2007.
    reprint

  46. On affine frames with transcendental dilations
    Proceedings of the American Mathematical Society, 135:211-216, 2007.
    reprint

  47. Affine systems that span Lebesgue spaces
    (With H.-Q. Bui.)
    Journal of Fourier Analysis and Applications, 11:533-556, 2005.
    reprint

    Papers 23 and 27 are based on the following research report:

    Spanning and sampling in Lebesgue and Sobolev spaces
    (With H.-Q. Bui.)
    University of Canterbury Research Report UCDMS2004/8, 64 pages.
    preprint

  48. New dissipated energies for the thin fluid film equation
    Communications on Pure and Applied Analysis, 4:613-634, 2005.
    reprint

  49. Another way to say subsolution: the maximum principle and sums of Green functions
    (With N. A. Watson.)
    Mathematica Scandinavica, 97:127-153, 2005.
    reprint

  50. Potential theory of the farthest-point distance function
    (With Igor E. Pritsker.)
    Canadian Mathematical Bulletin, 46:373-387, 2003.
    reprint

  51. Heteroclinic orbits, mobility parameters and stability for thin film type equations
    (With Mary C. Pugh.)
    Electronic Journal of Differential Equations, 2002: No. 95, 1-29.
    reprint

  52. Energy levels of steady states for thin film type equations
    (With Mary C. Pugh.)
    Journal of Differential Equations, 182:377-415, 2002.
    reprint

  53. A characterization of the higher dimensional groups associated with continuous wavelets
    (With Nik Weaver, Guido Weiss and Edward Wilson.)
    Journal of Geometric Analysis 12:89-102, 2002.
    reprint

  54. Translational averaging for completeness, characterization and oversampling of wavelets
    Collectanea Mathematica, 53:211-249, 2002.
    reprint

  55. Completeness of orthonormal wavelet systems, for arbitrary real dilations
    Applied and Computational Harmonic Analysis, 11:455-473, 2001.
    reprint

  56. Linear stability of steady states for thin film and Cahn-Hilliard type equations
    (With Mary C. Pugh.)
    Archive for Rational Mechanics and Analysis, 154:3-51, 2000.
    reprint

  57. Properties of steady states for thin film equations
    (With Mary C. Pugh.)
    European Journal of Applied Mathematics, 11(3):293-351, 2000.
    reprint

  58. Binary forms, equiangular polygons and harmonic measure
    (With Michael A. Bean.)
    Rocky Mountain Journal of Mathematics, 30:15-62, 2000.
    request reprint

  59. Eigenvalues of strings and cylinders with variable mass density
    Communications in Analysis and Geometry, 8:393-443, 2000.
    request reprint

  60. Eigenvalues of the Laplacian on inhomogeneous membranes
    American Journal of Mathematics, 120:305-344, 1998.
    request reprint

  61. Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
    Illinois Journal of Mathematics, 42:19-39, 1998.
    reprint

  62. Extremals for eigenvalues of Laplacians under conformal mapping
    (With Carlo Morpurgo.)
    Journal of Functional Analysis, 155:64-108, 1998.
    reprint

  63. Planar harmonic maps with inner and Blaschke dilatations
    Journal of the London Mathematical Society, 56:37-48, 1997.
    reprint

  64. Inequalities for the first eigenvalues of the clamped plate and buckling problems
    (With Mark S. Ashbaugh and Rafael D. Benguria.)
    International Series of Numerical Mathematics, Volume 123 (Proceedings of the Oberwolfach Conference "General Inequalities 7"), pp. 95-110, 1997.
    reprint

  65. Fundamental tones and buckling loads of clamped plates
    (With Mark S. Ashbaugh.)
    Annali della Scuola Normale Superiore di Pisa, 23:383-402, 1996.
    request reprint

  66. The argument principle for harmonic functions
    (With Peter L. Duren and Walter Hengartner.)
    American Mathematical Monthly, 103:411-415, 1996
    request reprint

  67. Injectivity can fail for higher-dimensional harmonic extensions
    Complex Variables, 28:357-369, 1996.
    request reprint

  68. Conformal mapping of long quadrilaterals and thick doubly connected domains
    Constructive Approximation, 10:523-554, 1994.
    request reprint

  69. Extremal problems involving logarithmic and Green capacity
    Duke Mathematical Journal, 70:445-480, 1993.
    request reprint

Support from the NSF, and the University of Illinois Research Board, the New Zealand Institute for Mathematics and its Applications, and the Visiting Erskine Fellowship at the University of Canterbury, is gratefully acknowledged.