Mathematics

Richard S. Laugesen - Research

Research Areas

1. Eigenvalues of differential operators, such as the Laplacian and bi-Laplacian.
These eigenvalues represent frequencies of vibration, rates of decay to equilibrium, quantum energy levels, and so on. The goal is to understand how the geometry of the domain influences the spectrum. I especially enjoy the challenge of proving sharp inequalities where the optimal domain possesses some symmetry e.g. disk, equilateral triangle.

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

Publications

Please e-mail me if you have trouble downloading the files you want.
You might enjoy this IMU report on impact factors and citation statitics.

38. Wavelets in Littlewood-Paley space, and Mexican hat completeness
    (With H.-Q. Bui.)
    14 pages. Submitted.
    preprint
    For related estimates, see our technical report A note on the solution of the Mexican hat problem.
    
    
37. Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality
    (With B. Siudeja.)
    20 pages. Submitted.
    preprint
    
    
36. Maximizing Neumann fundamental tones of triangles
    (With B. Siudeja.)
    22 pages. Journal of Mathematical Physics, to appear.
    preprint
    
    
35. Approximately dual frames in Hilbert spaces and applications to Gabor frames
    (With O. Christensen.)
    15 pages. Submitted.
    preprint
    
    
34. Frequency-scale frames and the solution of the Mexican hat problem
    (With H.-Q. Bui.)
    26 pages. Submitted.
    preprint
    
    
33. Gabor dual spline windows
    Applied and Computational Harmonic Analysis, 27:180-194, 2009.
    reprint, or an earlier preprint with more details
    
    
32. 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
    
    
31. Moment inequalities for equilibrium measures in the plane
    (With A. Baernstein II and I. E. Pritsker.)
    Pure and Applied Mathematics Quarterly, 7:47-82, 2011, to appear.
    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.