Fixed frequency eigenfunction immersions and supremum norms of random waves

Previous Article
Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result
ERA-MS Home
This Volume
Next Article
A Besicovitch cylindrical transformation with Hölder function

January 2015, 22: 76-86. doi: 10.3934/era.2015.22.76

Fixed frequency eigenfunction immersions and supremum norms of random waves

1.	Department of Mathematics, Harvard University, Cambridge, United States
2.	Department of Mathematics, Massachusetts Institute of Technology, Cambridge, United States

Received January 2015 Revised June 2015 Published September 2015

Abstract
Related Papers

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.

Keywords: Immersions by eigenfunctions, supremum norms, random waves..

Mathematics Subject Classification: 35P05 (Pri), 53C42 (Sec.

Citation: Yaiza Canzani, Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of random waves. Electronic Research Announcements, 2015, 22: 76-86. doi: 10.3934/era.2015.22.76

References:

[1]	R. Adler and J. Taylor, Random Fields and Geometry,, Springer Monographs in Mathematics, (2007). Google Scholar
[2]	A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425. doi: 10.1090/S0002-9947-08-04456-5. Google Scholar
[3]	P. Bartlett, Theoretical statistics,, lecture 14, (). Google Scholar
[4]	N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917. Google Scholar
[5]	Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, (). Google Scholar
[6]	J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. doi: 10.1007/BF01405172. Google Scholar
[7]	R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields,, J. Func. Anal., 266 (2014), 5085. doi: 10.1016/j.jfa.2014.02.012. Google Scholar
[8]	L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193. doi: 10.1007/BF02391913. Google Scholar
[9]	M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves,, Ann. of Math. (2), 177 (2013), 699. doi: 10.4007/annals.2013.177.2.8. Google Scholar
[10]	F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus,, Ann. Inst. Fourier (Grenoble), 58 (2008), 299. doi: 10.5802/aif.2351. Google Scholar
[11]	V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary,, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25. Google Scholar
[12]	J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35. doi: 10.1524/stnd.2008.0911. Google Scholar
[13]	L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037. doi: 10.1512/iumj.2014.63.5321. Google Scholar
[14]	J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000). Google Scholar
[15]	B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627. Google Scholar
[16]	G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99. Google Scholar
[17]	N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008. Google Scholar
[18]	J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329. doi: 10.1155/S1073792897000238. Google Scholar
[19]	S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321. doi: 10.1090/conm/484/09482. Google Scholar
[20]	S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415. doi: 10.1006/jfan.1996.2981. Google Scholar
[21]	S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317. doi: 10.1155/S107379289800021X. Google Scholar

show all references

References:

[1]	R. Adler and J. Taylor, Random Fields and Geometry,, Springer Monographs in Mathematics, (2007). Google Scholar
[2]	A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425. doi: 10.1090/S0002-9947-08-04456-5. Google Scholar
[3]	P. Bartlett, Theoretical statistics,, lecture 14, (). Google Scholar
[4]	N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917. Google Scholar
[5]	Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, (). Google Scholar
[6]	J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. doi: 10.1007/BF01405172. Google Scholar
[7]	R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields,, J. Func. Anal., 266 (2014), 5085. doi: 10.1016/j.jfa.2014.02.012. Google Scholar
[8]	L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193. doi: 10.1007/BF02391913. Google Scholar
[9]	M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves,, Ann. of Math. (2), 177 (2013), 699. doi: 10.4007/annals.2013.177.2.8. Google Scholar
[10]	F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus,, Ann. Inst. Fourier (Grenoble), 58 (2008), 299. doi: 10.5802/aif.2351. Google Scholar
[11]	V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary,, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25. Google Scholar
[12]	J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35. doi: 10.1524/stnd.2008.0911. Google Scholar
[13]	L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037. doi: 10.1512/iumj.2014.63.5321. Google Scholar
[14]	J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000). Google Scholar
[15]	B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627. Google Scholar
[16]	G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99. Google Scholar
[17]	N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008. Google Scholar
[18]	J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329. doi: 10.1155/S1073792897000238. Google Scholar
[19]	S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321. doi: 10.1090/conm/484/09482. Google Scholar
[20]	S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415. doi: 10.1006/jfan.1996.2981. Google Scholar
[21]	S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317. doi: 10.1155/S107379289800021X. Google Scholar

[1]	Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[2]	Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[3]	Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[4]	Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[5]	Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[6]	Daniel T. Wise. Nonpositive immersions, sectional curvature, and subgroup properties. Electronic Research Announcements, 2003, 9: 1-9.
[7]	Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671
[8]	Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
[9]	Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[10]	Giovanni Bellettini, Matteo Novaga, Shokhrukh Yusufovich Kholmatov. Minimizers of anisotropic perimeters with cylindrical norms. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1427-1454. doi: 10.3934/cpaa.2017068
[11]	Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[12]	Guojing Zhang, Steve Rosencrans, Xuefeng Wang, Kaijun Zhang. Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1061-1079. doi: 10.3934/dcds.2009.25.1061
[13]	Tohru Wakasa, Shoji Yotsutani. Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2010, 9 (2) : 539-561. doi: 10.3934/cpaa.2010.9.539
[14]	Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks & Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239
[15]	Haïm Brezis. Remarks on some minimization problems associated with BV norms. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-17. doi: 10.3934/dcds.2019242
[16]	Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[17]	Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729
[18]	Guanglu Zhou. A quadratically convergent method for minimizing a sum of Euclidean norms with linear constraints. Journal of Industrial & Management Optimization, 2007, 3 (4) : 655-670. doi: 10.3934/jimo.2007.3.655
[19]	Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[20]	Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure & Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341

2018 Impact Factor: 0.263

American Institute of Mathematical Sciences