Mathematics

Richard S. Laugesen - Research

Research Areas

1. Eigenvalues of differential operators (esp. Laplacian and bi-Laplacian)
Eigenvalues represent 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 in 2010. Here is an overview of 2000 years of work on isoperimetric problems in mathematical physics.

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 (L^p, Sobolev, Hardy)?

Publications

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You might enjoy this IMU report on impact factors and citation statitics.

48. Multivariable feedback particle filter
    (With T. Yang, P. G. Mehta and S. Meyn.)
    8 pages. Submitted.
    preprint request
    
    
47. Wavelet frame bijectivity on Lebesgue and Hardy spaces
    (With H.-Q. Bui.)
    24 pages. Submitted.
    preprint
    
    
46. Rebuttal of Donnelly's paper on the spectral gap
    (With M. S. Ashbaugh and A. Henrot.)
    Mathematische Zeitschrift 269 (2011), 5-7.
    reprint
    
    
45. Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian
    21 pages. Proceedings of the AMSI International Conference on Harmonic Analysis and Applications (Macquarie University, 2011), to appear.
    preprint
    
    
44. Sums of magnetic eigenvalues are maximal on rotationally symmetric domains
    (With J. Liang and A. Roy.)
    Annales Henri Poincare 13 (2012), 731-750.
    reprint
    
    
43. Uniqueness for the continuous wavelet transform
    (With H.-Q. Bui.)
    Far East Journal of Applied Mathematics 65 (2012), 1-11.
    reprint
    
    
42. Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals
    (With Z. C. Pan and S. S. Son.)
    14 pages. Mathematical Inequalities and Applications, to appear.
    reprint (in forthcoming papers) or arXiv preprint
    
    
41. 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).
    
    
40. 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
    
    
39. Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane
    (With B. A. Siudeja.)
    Journal of Functional Analysis 260 (2011), 1795-1823.
    reprint
    
    
38. 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
    
    
37. 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.
    
    
36. 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
    
    
34. Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality
    (With B. A. Siudeja.)
    Journal of Differential Equations 249 (2010), 118-135.
    reprint
    
    
33. Maximizing Neumann fundamental tones of triangles
    (With B. A. Siudeja.)
    22 pages. Journal of Mathematical Physics, 50:112903, 2009.
    reprint
    
    
32. Gabor dual spline windows
    Applied and Computational Harmonic Analysis, 27:180-194, 2009.
    reprint, or an earlier preprint with more details
    
    
31. 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
    
    
30. 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
    
    
29. Affine synthesis onto L^p when 0 < p <= 1
    Journal of Fourier Analysis and Applications, 14:235-266, 2008.
    reprint
    
    
28. 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
    
    
27. Sobolev spaces and approximation by affine spanning systems
    (With H.-Q. Bui.)
    Mathematische Annalen, 341:347-389, 2008.
    reprint
    
    
26. Approximation and spanning in the Hardy space, by affine systems
    (With H.-Q. Bui.)
    Constructive Approximation, 28:149-172, 2008.
    reprint
    
    
25. On convex surfaces with minimal moment of inertia
    (With P. Freitas and G. F. Liddell.)
    Journal of Mathematical Physics, 48, 122902, 2007.
    reprint
    
    
24. On affine frames with transcendental dilations
    Proceedings of the American Mathematical Society, 135:211-216, 2007.
    reprint
    
    
23. 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
    
    
22. New dissipated energies for the thin fluid film equation
    Communications on Pure and Applied Analysis, 4:613-634, 2005.
    reprint
    
    
21. Another way to say subsolution: the maximum principle and sums of Green functions
    (With N. A. Watson.)
    Mathematica Scandinavica, 97:127-153, 2005.
    reprint
    
    
20. Potential theory of the farthest-point distance function
    (With Igor E. Pritsker.)
    Canadian Mathematical Bulletin, 46:373-387, 2003.
    reprint
    
    
19. 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
    
    
18. Energy levels of steady states for thin film type equations
    (With Mary C. Pugh.)
    Journal of Differential Equations, 182:377-415, 2002.
    reprint
    
    
17. 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
    
    
16. Translational averaging for completeness, characterization and oversampling of wavelets
    Collectanea Mathematica, 53:211-249, 2002.
    reprint
    
    
15. Completeness of orthonormal wavelet systems, for arbitrary real dilations
    Applied and Computational Harmonic Analysis, 11:455-473, 2001.
    reprint
    
    
14. 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
    
    
13. Properties of steady states for thin film equations
    (With Mary C. Pugh.)
    European Journal of Applied Mathematics, 11(3):293-351, 2000.
    reprint
    
    
12. Binary forms, equiangular polygons and harmonic measure
    (With Michael A. Bean.)
    Rocky Mountain Journal of Mathematics, 30:15-62, 2000.
    request reprint
    
    
11. Eigenvalues of strings and cylinders with variable mass density
    Communications in Analysis and Geometry, 8:393-443, 2000.
    request reprint
    
    
10. Eigenvalues of the Laplacian on inhomogeneous membranes
    American Journal of Mathematics, 120:305-344, 1998.
    request reprint
    
    
9. Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
   Illinois Journal of Mathematics, 42:19-39, 1998.
   reprint
   
   
8. Extremals for eigenvalues of Laplacians under conformal mapping
   (With Carlo Morpurgo.)
   Journal of Functional Analysis, 155:64-108, 1998.
   reprint
   
   
7. Planar harmonic maps with inner and Blaschke dilatations
   Journal of the London Mathematical Society, 56:37-48, 1997.
   reprint
   
   
6. 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
   
   
5. 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
   
   
4. The argument principle for harmonic functions
   (With Peter L. Duren and Walter Hengartner.)
   American Mathematical Monthly, 103:411-415, 1996
   request reprint
   
   
3. Injectivity can fail for higher-dimensional harmonic extensions
   Complex Variables, 28:357-369, 1996.
   request reprint
   
   
2. Conformal mapping of long quadrilaterals and thick doubly connected domains
   Constructive Approximation, 10:523-554, 1994.
   request reprint
   
   
1. 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.