Norman Levenberg

Uniform approximation by discrete least squares polynomials

We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these polynomials provide nearly optimal approximants. For analytic functions, similar results may be achieved on more general K by allowing the number of points to grow at a slightly larger rate.

On the spacing of Fekete points for a sphere, ball or simplex

Suppose that K ⊂ ℝd is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c1, c2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, Fn ⊂ K, Download full-size image for all a ∈ Fn. Here dist(a, b) is a natural distance on K that will be described in the text.

Quasianalyticity and pluripolarity

be the graph of / in C2. The set Tf is always pluripolar when / is a real analytic function. In [DF], Diederich and Fornaess give an example of a C?? function / with nonpluripolar graph in C2. The paper [LMP] contains an example of a holomorphic function / on the unit disk U, continuous up to the boundary, such that the graph of / over S is not pluripolar as a subset of C2. Thus a priori the pluripolarity of graphs of functions on S is indeterminate. In this paper we prove that graphs of quasianalytic functions are still pluripolar (all necessary definitions can be found in the next section). More precisely,

Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

We use geometric methods to calculate a formula for the complex Monge-Ampere measure (dd c V K ) n , for K ⋐ ℝ n C ℂ n a convex body and V K its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex Monge-Ampere solution V K .

Pseudometrics, distances and multivariate polynomial inequalities

We discuss three natural pseudodistances and pseudometrics on a bounded domain in R^N based on polynomial inequalities.

On the Siciak extremal function for real compact convex sets

is the class of plurisubharmonic functions of logarithmic growth (here we have Izl=(L~=1IzjI2)1/2 and log+ Izl=max{O,loglzl}). Then the upper semicontinuous regularization VE(z):=limsuPC-+z VE(() is called the (Siciak) extremal function of E. If K is a compact set in eN, then the extremal function in (1.1) can be gotten via the formula (1.2) VK(Z) :=max{ o,suP{ de~p log Ip(z)l:p holomorphic polynomial, IIpIIK:::; 1} } (Theorem 5.1.7 in [KI]). Here, IlpIlK:=SUPzEKIp(z)1 denotes the uniform norm on K. We say that K is regular if and only if V;=VK. Note that if we let

Extremal functions for real convex bodies

We study the smoothness of the Siciak–Zaharjuta extremal function associated to a convex body in

Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points

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Multivariate simultaneous approximation

. We also prove a formula relating the complex equilibrium measure of a convex body in

Polynomial interpolation and approximation in

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