Abstract
Let {φ k } be an orthonormal system on a quasi-metric measure space \({\mathbb{X}}\), {ℓ k } be a nondecreasing sequence of numbers with lim k→∞ ℓ k =∞. A diffusion polynomial of degree L is an element of the span of {φ k :ℓ k ≤L}. The heat kernel is defined formally by \(K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}\). If T is a (differential) operator, and both K t and T y K t have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖ p ≤c 1 L c‖P‖ p . In particular, we are interested in the case when \({\mathbb{X}}\) is a Riemannian manifold, T is a derivative operator, and \(p\not=2\). In the case when \({\mathbb{X}}\) is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion polynomial valued operators.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belkin, M., Niyogi, P.: Semi-supervised learning on Riemannian manifolds. Mach. Learn. 56, 209–239 (2004)
Belkin, M., Matveeva, I., Niyogi, P.: Regularization and regression on large graphs. In: Proc. of Computational Learning Theory, Banff, Canada (2004)
Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, San Diego (2003)
Cheng, S.Y., Li, P., Yau, S.-T.: On the upper estimate of the heat kernel of a complete Riemannian manifold. Am. J. Math. 103(5), 1021–1063 (1981)
Chui, C.K., Donoho, D.L.: Special Issue: Diffusion maps and wavelets. Appl. Comput. Harmon. Anal. 21(1) (2006)
Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53–94 (2006)
Davies, E.B.: L p spectral theory of higher-order elliptic differential operators. Bull. Lond. Math. Soc. 29, 513–546 (1997)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New York (1976)
do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)
Donoho, D.L., Grimes, C.: Image manifolds which are isometric to Euclidean space. http://www-stat.stanford.edu/~donoho/Reports/2002/WhenIsometry.pdf
Donoho, D.L., Levi, O., Starck, J.-L., Martinez, V.J.: Multiscale geometric analysis for 3-D catalogues. http://www-stat.stanford.edu/~donoho/Reports/2002/MGA3D.pdf
Dungey, N.: Some remarks on gradient estimates for heat kernels. Abstr. Appl. Anal. 2006, Art. ID 73020, 10 pp.
Filbir, F., Themistoclakis, W.: Polynomial approximation on the sphere using scattered data. Math. Nachr. 281(5), 650–668 (2008)
Grigor’yan, A.: Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold. J. Funct. Anal. 127(2), 363–389 (1995)
Grigor’yan, A.: Estimates of heat kernels on Riemannian manifolds. In: Davies, E.B., Safarov, Y. (eds.) Spectral Theory and Geometry, Edinburgh, 1998. London Math. Soc. Lecture Note Ser., vol. 273, pp. 140–225. Cambridge University Press, Cambridge (1999)
Grigor’yan, A.: Heat kernels and function theory on metric measure spaces. Contemp. Math. 338, 143–172 (2003)
He, X., Yan, S., Hu, Y., Niyogi, P., Zhang, H.-J.: Face recognition using laplacianfaces. IEEE Trans. Pattern Anal. Mach. Intell. 27(3), 328–340 (2005)
Jones, P.W., Maggioni, M., Schul, R.: Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian. Manuscript, arXiv:0709.1975
Keiner, J., Kunis, S., Potts, D.: Efficient reconstruction of functions on the sphere from scattered data. J. Fourier Anal. Appl. 13, 435–458 (2007)
Kordyukov, Yu.A.: Lp-theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23(3), 223–260 (1991)
Korevaar, J.: Tauberian theory: A century of developments. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 329. Springer, Berlin (2004)
Korevaar, J.: A Tauberian boundedness theorem of the Ikehara type. Manuscript
Le Gia, Q.T., Mhaskar, H.N.: Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Numer. Anal. 47(1), 440–466 (2008)
Maggioni, M., Mhaskar, H.N.: Diffusion polynomial frames on metric measure spaces. Appl. Comput. Harmon. Anal. 24(3), 329–353 (2008)
Mhaskar, H.N.: Approximation theory and neural networks. In: Jain, P.K., Krishnan, M., Mhaskar, H.N., Prestin, J., Singh, D. (eds.) Wavelet Analysis and Applications, Proceedings of the International Workshop in Delhi, 1999, pp. 247–289. Narosa Publishing, New Delhi (2001)
Mhaskar, H.N.: Polynomial operators and local smoothness classes on the unit interval, II. Jaén J. Approx. 1(1), 1–25 (2009)
Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Chui, C.K., Neamtu, M., Schumaker, L. (eds.) Approximation Theory XI, Gatlinburg, 2004, pp. 287–318. Nashboro Press, Brentwood (2005)
Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comput. 70(235), 1113–1130 (2001). Corrigendum: Math. Comput. 71, 453–454 (2001)
Minakshisundaram, S.: Eigenfunctions on Riemannian manifolds. J. Indian Math. Soc. (N.S.) 17, 159–165 (1953–1954)
Nikolskii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York (1975)
Oneill, B.: Elementary Differential Geometry. Elsevier, Amsterdam (2006)
Pesenson, I.: Bernstein-Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds. J. Approx. Theory 150(2), 175–198 (2008)
Rivlin, T.J.: Chebyshev Polynomials. Wiley, New York (1990)
Schmid, D.: Scattered data approximation on the rotation group and generalizations. Ph.D. thesis, Technische Universität München (2009)
Sikora, A.: Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247, 643–662 (2004)
Taylor, M.E.: Partial Differential Equations, Basic Theory. Springer, New York (1996)
Xu, B.: Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold. Ann. Glob. Anal. Geom. 26, 231–252 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Karlheinz Gröechenig.
The research of F. Filbir was partially funded by Deutsche Forschungsgemeinschaft grant FI 883/3-1 and PO711/9-1.
The research of H.N. Mhaskar was supported, in part, by grant DMS-0605209 and its continuation DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the US Army Research Office.
Rights and permissions
About this article
Cite this article
Filbir, F., Mhaskar, H.N. A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel. J Fourier Anal Appl 16, 629–657 (2010). https://doi.org/10.1007/s00041-010-9119-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-010-9119-4
Keywords
- Approximation on manifolds
- Bernstein inequalities
- Marcinkiewicz-Zygmund inequalities
- Quadrature formulas