Discrepancy Theorems Research Articles

OPTIMAL SPLITTING OF PARSEVAL FRAMES USING WALSH MATRICES

In 2014 Adam Marcus, Daniel Spielman and Nikhil Srivastava used random vectors to prove a key discrepancy theorem and in so doing gave a positive answer to the long-standing Kadison-Singer Problem. In this paper we use Walsh matrices to construct a class of natural frames in Euclidean space and discuss how these frames relate to the key discrepancy theorem.

Discrepancy Theorems for the Zero Distribution of Rational Functions

In the theory of zero distribution of polynomials, there are two branches that are closely related to each other. One branch deals with the conditions under which the normalized zero counting measure of a sequence of polynomials converges in the weak-star sense to the equilibrium measure of a Jordan curve. The other consists of discrepancy theorems, where the maximum deviation of these two measures is estimated. There are also theorems on weak-star convergence of the zero distribution of special sequences of rational functions to a measure that depends on the poles of these functions. In this paper, we give appropriate discrepancy theorems for rational functions. These new results also imply theorems about the weak-star convergence of the zero distribution of more general sequences of rational functions.

Erdős-Turʼan-Type Discrepancy for the Zeros of Rational Functions

Abstract In 1950 P. Erdős and P. Turan published a discrepancy theorem for the zeros of a polynomial. Therein, the maximum deviation of the normalized zero counting measure from the equilibrium measure of the unit circle is estimated. Many other discrepancy theorems and related propositions about weak-star-convergence of the zero distribution of a sequence of polynomials were proved during the last decades. For several years the weak-star-convergence of the zero distribution of a sequence of rational functions is also studied. The main result of this paper is a discrepancy theorem for the zero distribution of a rational function which generalizes and sharpens previous propositions about weak-star-convergence of the zero counting measure of sequences of rational functions and known discrepancy theorems for polynomials.

On zeros of polynomials orthogonal over a convex domain

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.

A local discrepancy theorem

Let σ be a signed measure on [−1,1], with total mass 0, and [ a,b] ⊆ [−1,1]. We estimate |σ([ a,b])| in terms of the logarithmic potential of σ in a complex neighborhood of [ a,b]. Applications include a detailed description of the distribution of zeros of a class of orthogonal polynomials, including Pollaczek polynomials.

A discrepancy theorem for dini-smooth arcs

If L is a Dini-smooth arc in the complex plane we obtain estimates for the discrepancy between the equilibrium measure μ = μLof L and an arbitrary positive Borel unit measure v supported on L based on one-sided bounds for the-difference of their logarithmic potentials along level lines of the conformal mapping of onto the exterior of unit disk with standard normalization at ∞.

Some Discrepancy Theorems in the Theory of Weighted Polynomial Approximation

We obtain discrepancy theorems for the distribution of the zeros of extremal polynomials arising in the theory of weighted polynomial approximation on the whole real axis.

A Discrepancy Theorem on Quasiconformal Curves

In [7], Blatt and Mhaskar estimated the Erdős-Turan type discrepancy of a signed Borel measure σ on a sufficiently smooth Jordan curve or arc L in terms of the logarithmic potential of σ on a curve enclosing L. We extend this result to a measure σ on an arbitrary quasiconformal curve. As applications, estimates for the distribution of simple zeros of monic polynomials, Fekete points, extreme points of polynomials of best uniform approximation are obtained.

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1

Letw be a suitable weight function,Bn,p denote the polynomial of best approximation to a functionf inLwp[−1, 1],vn be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofBn+1,p−Bn,p and μ be the arcsine measure defined by\(d\mu : = (\pi \sqrt {1 - x^2 } )^{ - 1} dx\). We estimate the rate at which the sequencevn converges to μ in the weak-* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalLwp norm.

On discrepancy theorems with applications to approximation theory

We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turan on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

A general discrepancy theorem

A general discrepancy theorem