Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f\u27 is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval

Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the $L_\infty$ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. We present a new approach based on distortion of the coefficients in an $L_q$ norm with $q<\infty$ which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis.Comment: 24 page

The Averaging Lemma

Averaging lemmas deduce smoothess of velocity averages, such as f(x) := Z f(x; v) dv; IR d ; from properties of f . A canonical example is that f is in the Sobolev space W 1=2 (L 2 (IR d )) whenever f and g(x; v) := v r x f(x; v) are in L 2 (IR d 4 The present paper shows how techniques from Harmonic Analysis such as maximal functions wavelet decompositions and interpolation can be used to prove L p versions of the averaging lemma. For example, it is shown that f; g 2 L p (IR d implies that f is in the Besov space B s p (L p (IR d )), s := min(1=p; 1=p 0 ). Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint p = 1. AMS subject classication: 35L60, 35L65, 35B65, 46B70, 46B45, 42B25. Key Words: averaging lemma, regularity, transport equations, Besov spaces 1 Introduction Averaging lemmas arise in the study of regularity of solut..

Degree of Adaptive Approximation

We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation

Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Mat\'ern kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds.Comment: 29 pages. To appear in Festschrift for the 80th Birthday of Ian Sloa

Interpolation of Besov-Spaces

We investigate Besov spaces and their connection with dyadic spline approximation in Lp(Omega), 0 \u3c p (less than or equal to) infinity. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

Approximation and learning by greedy algorithms

We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118--2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.Comment: Published in at http://dx.doi.org/10.1214/009053607000000631 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Error-bounds for Gaussian Quadrature and Weighted-L1 Polynomial Approximation

Error bounds for Gaussian quadrature are given in terms of the number of quadrature points and smoothness properties of the function whose integral is being approximated. An intermediate step involves a weighted-L\u27 polynomial approximation problem which is treated in a more general context than that specifically required to bound the Gaussian quadrature error