## 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 (L^{p}, Sobolev,
Hardy)?

#### Publications

- Many of my recent papers can be found here on the ArXiv.
- You might enjoy this IMU report on impact factors and citation statitics.

**Robin spectrum: two disks maximize the third eigenvalue**

(With A. Girouard.)

Submitted. 32 pages.

preprint

**The Robin Laplacian --- spectral conjectures, rectangular theorems**

Submitted. 44 pages.

preprint (correction: Conjecture D is now known to be false)

**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

**Symmetrization in Analysis**

(By Albert Baernstein II, with David Drasin and Richard Laugesen.)

Cambridge University Press, 2019. 474 pages.

purchase

**From Steklov to Neumann and beyond, via Robin: the Szegő way**

(With P. Freitas.)

Canadian Journal of Mathematics. 20 pages.

reprint

**From Neumann to Steklov and beyond, via Robin: the Weinberger way**

(With P. Freitas.)

Submitted. 23 pages.

preprint

**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

**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.)

**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

**Shifted lattices and asymptotically optimal ellipses**

(With S. Liu.)

The Journal of Analysis, 26 (2018) no. 1, 71-102.

reprint

**Optimal stretching for lattice points and eigenvalues**

(With S. Liu.)

Arkiv för Matematik, 56 (2018) no. 1, 111-145.

reprint

**Optimal stretching for lattice points under convex curves**

(With S. Ariturk.)

Portugaliae Mathematica (N.S.), 74 (2017) no. 2, 91-114.

reprint

**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

**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

**Preparing graduates for careers in the mathematical sciences**

(With S. Minkoff, W. Menasco, F. Santosa, S. Pankavich.)

SIAM News, 49 (2016).

reprint

**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

**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

**Multivariable feedback particle filter**

(With T. Yang, P. G. Mehta and S. Meyn.)

Automatica, 71 (2016), 10-23.

reprint

**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

**Magnetic spectral bounds on starlike plane domains**

(With B. A. Siudeja.)

ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 670-689.

reprint

**Sharp spectral bounds on starlike domains**

(With B. A. Siudeja.)

Journal of Spectral Theory, 4 (2014), 309-347.

reprint

**Explicit interpolation bounds between Hardy space and L**^{2}

(With H.-Q. Bui.)

Journal of the Australian Mathematical Society, 95 (2013), 158-168.

reprint

**Wavelet frame bijectivity on Lebesgue and Hardy spaces**

(With H.-Q. Bui.)

Journal of Fourier Analysis and Applications 19 (2013), 376-409.

reprint

**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

**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

**Uniqueness for the continuous wavelet transform**

(With H.-Q. Bui.)

Far East Journal of Applied Mathematics 65 (2012), 1-11.

reprint

**Sums of magnetic eigenvalues are maximal on rotationally symmetric domains**

(With J. Liang and A. Roy.)

Annales Henri Poincare 13 (2012), 731-750.

reprint

**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

**Rebuttal of Donnelly's paper on the spectral gap**

(With M. S. Ashbaugh and A. Henrot.)

Mathematische Zeitschrift 269 (2011), 5-7.

reprint

**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).

**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

**Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane**

(With B. A. Siudeja.)

Journal of Functional Analysis 260 (2011), 1795-1823.

reprint

**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

**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.

**Frequency-scale frames and the solution of the Mexican hat problem**

(With H.-Q. Bui.)

Constructive Approximation 33 (2011), 163–189.

reprint

**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

**Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality**

(With B. A. Siudeja.)

Journal of Differential Equations 249 (2010), 118-135.

reprint

**Maximizing Neumann fundamental tones of triangles**

(With B. A. Siudeja.)

22 pages. Journal of Mathematical Physics,**50**:112903, 2009.

reprint

**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

**A note on constructing affine systems for L**^{2}

(With H.-Q. Bui and N. Kaiblinger.)

Applied and Computational Harmonic Analysis, 25:400-406, 2008.

reprint

**Affine synthesis onto L**^{p}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.