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 (Lp, Sobolev, Hardy).
Publications
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if you have trouble downloading the files you want.
You might enjoy this IMU report on
impact factors and citation statitics.
- 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.
- Minimizing Neumann fundamental tones of triangles:
an optimal Poincare inequality
(With B. Siudeja.)
20 pages. Submitted.
preprint
- Maximizing Neumann fundamental tones of triangles
(With B. Siudeja.)
22 pages. Journal of Mathematical Physics, to appear.
preprint
- Approximately dual frames in Hilbert spaces and
applications to Gabor frames
(With O. Christensen.)
15 pages. Submitted.
preprint
- Frequency-scale frames and the solution of the
Mexican hat problem
(With H.-Q. Bui.)
26 pages. Submitted.
preprint
- Gabor dual spline windows
Applied and Computational Harmonic Analysis, 27:180-194, 2009.
reprint, or an earlier preprint with more details
- 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
- 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
- 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
- Affine synthesis onto Lp when 0 < p <= 1
Journal of Fourier Analysis and Applications, 14:235-266, 2008.
reprint
- 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
- Sobolev spaces and approximation by affine spanning
systems
(With H.-Q. Bui.)
Mathematische Annalen, 341:347-389, 2008.
reprint
- Approximation and spanning in the Hardy space, by affine
systems
(With H.-Q. Bui.)
Constructive Approximation, 28:149-172, 2008.
reprint
- On convex surfaces with minimal moment of inertia
(With P. Freitas and G. F. Liddell.)
Journal of Mathematical Physics, 48, 122902, 2007.
reprint
- On affine frames with transcendental dilations
Proceedings of the American Mathematical Society, 135:211-216, 2007.
reprint
- 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
- New dissipated energies for the thin fluid film equation
Communications on Pure and Applied Analysis, 4:613-634, 2005.
reprint
- Another way to say subsolution: the maximum principle and
sums of Green functions
(With N. A. Watson.)
Mathematica Scandinavica, 97:127-153, 2005.
reprint
- Potential theory of the farthest-point distance function
(With Igor E. Pritsker.)
Canadian Mathematical Bulletin, 46:373-387, 2003.
reprint
- 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
- Energy levels of steady states for thin film type equations
(With Mary C. Pugh.)
Journal of Differential Equations, 182:377-415, 2002.
reprint
- 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
- Translational averaging for completeness, characterization and
oversampling of wavelets
Collectanea Mathematica, 53:211-249, 2002.
reprint
- Completeness of orthonormal wavelet systems, for arbitrary
real dilations
Applied and Computational Harmonic Analysis, 11:455-473, 2001.
reprint
- 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
- Properties of steady states for thin film equations
(With Mary C. Pugh.)
European Journal of Applied Mathematics, 11(3):293-351, 2000.
reprint
- Binary forms, equiangular polygons and harmonic measure
(With Michael A. Bean.)
Rocky Mountain Journal of Mathematics, 30:15-62, 2000.
request reprint
- Eigenvalues of strings and cylinders with variable mass
density
Communications in Analysis and Geometry, 8:393-443, 2000.
request reprint
- Eigenvalues of the Laplacian on inhomogeneous membranes
American Journal of Mathematics, 120:305-344, 1998.
request reprint
- Eigenvalues of Laplacians with mixed boundary conditions,
under conformal mapping
Illinois Journal of Mathematics, 42:19-39, 1998.
reprint
- Extremals for eigenvalues of Laplacians under conformal
mapping
(With Carlo Morpurgo.)
Journal of Functional Analysis, 155:64-108, 1998.
reprint
- Planar harmonic maps with inner and Blaschke dilatations
Journal of the London Mathematical Society, 56:37-48, 1997.
reprint
- 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
- 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
- The argument principle for harmonic functions
(With Peter L. Duren and Walter Hengartner.)
American Mathematical Monthly, 103:411-415, 1996
request reprint
- Injectivity can fail for higher-dimensional harmonic
extensions
Complex Variables, 28:357-369, 1996.
request reprint
- Conformal mapping of long quadrilaterals and thick doubly
connected
domains
Constructive Approximation, 10:523-554, 1994.
request reprint
- 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.