A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel F. Filbir * , H. N. Mhaskar † Abstract Let {φk} be an orthonormal system on a quasi-metric measure space X, {ℓk} be a nondecreasing se- quence of numbers with limk→∞ ℓk = ∞. A diffusion polynomial of degree L is an element of the span of {φk : ℓk ≤ L}. The heat kernel is defined formally by Kt(x, y)= P ∞ k=0 exp(−ℓ 2 k t)φk(x) φk(y). If T is a (differential) operator, and both Kt and Ty Kt have Gaussian upper bounds, we prove the Bern- stein inequality: for every p,1 ≤ p ≤∞ and diffusion polynomial P of degree L, ‖TP ‖p ≤ c1L c ‖P ‖p. In particular, we are interested in the case when X is a Riemannian manifold, T is a derivative op- erator, and p = 2. In the case when 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. 1 Introduction Many practical applications, for example, document analysis [8], face recognition [17], semi–supervised learning [2, 1], image processing [10], and cataloguing of galaxies [11], involve a large amount of very high dimensional data. Typically, this data has a lower intrinsic dimensionality; for example, one may assume that it belongs to a low dimensional manifold in a high dimensional, ambient Euclidean space. The desire to take advantage of this low intrinsic dimensionality has recently prompted a great deal of research on diffusion geometry techniques. The special issue [7] of Applied and Computational Harmonic Analysis contains several papers that serve as a good introduction to this subject. An essential ingredient in these works is a data–dependent heat kernel K t on the manifold X in question, which can be defined formally by K t (x,y)=  k≥0 exp(−ℓ 2 k t)φ k (x) φ k (y) , t> 0, x,y ∈ X, where the eigenfunctions {φ k } are an orthonormal basis for L 2 (X,µ) for an appropriate measure µ, and ℓ k ’s are nonnegative numbers; the eigenvalues of the (square root of the negative) Laplacian. A multiresolution analysis is then defined by Coifman and Maggioni [8] for a fixed ǫ> 0 by defining the increasing sequence of scaling spaces span {φ k : exp(−2 −j ℓ 2 k ) ≥ ǫ} = span {φ k : ℓ 2 k ≤ (2 j log(1/ǫ))}. The range of the operators generated by K 2 −j being “close” to the space at level j , one may obtain an approximate projection of a function by applying these operators to the function. In turn, these * Institute of Biomathematics and Biometry, Helmholtz Center Munich, 85764 Neuherberg, Germany, email: filbir@helmholtz-muenchen.de. The research of this author was partially funded by Deutsche Forschungsgemeinschaft grant FI 883/3-1 and PO711/9-1. † Department of Mathematics, California State University, Los Angeles, California, 90032, USA, email: hmhaska@calstatela.edu. The research of this author was supported, in part, by grants from the National Science Foundation and the U.S. Army Research Office. 1