Hans-Peter Blatt

Discrepancy of signed measures and polynomial approximation

Auxiliary Facts.- Zero Distribution of Polynomials.- Discrepancy Theorems via Two-Sided Bounds for Potentials.- Discrepancy Thoerems via One-Sided Bounds for Potentials.- Discrepancy Theorems via Energy Integrals.- Applications of Jentzsch-Szego and Erdos-Turan Type Theorems.- Applications of Discrepancy Theorems.- Special Topics.- Appendix A: Conformally Invariant Characteristics of Curve Families.- Appendix B: Basics in the Theory of Quasiconformal Mappings.- Appendix C: Constructive Theory of Functions of a Complex Variable.- Appendix D: Miscellaneous Topics.- Bibliography.- Glossary of Notation.- Index.

Behavior of zeros of polynomials of near best approximation

Abstract The purpose of this paper is to study the asymptotic behavior of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example, suppose E is a finite union of compact intervals of the real line and f is a continuous function on E , but is not analytic on E ; then we show (cf. Corollary 2.2) that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E . This fact answers a question posed by P. Borwein who showed that, for the case when E is a single interval and f is real-valued, then the above hypotheses on f imply that at least one point of E is the limit point of zeros of such polynomials.

On the distribution of simple zeros of polynomials

Abstract Erdős and Turan discussed in ( Ann. of Math. 41 (1940), 162–173; 51 (1950), 105–119) the distribution of the zeros of monic polynomials if their Chebyshev norm on [−1, 1] or on the unit disk is known. We sharpen this result to the case that all zeros of the polynomials are simple. As applications, estimates for the distribution of the zeros of orthogonal polynomials and the distribution of the alternation points in Chebyshev polynomial approximation are given. This last result sharpens a well-known error bound of Kadec ( Amer. Math. Soc. Transl. 26 (1963), 231–234).

The distribution of extreme points in best complex polynomial approximation

AbstractLetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if

Rationale Tschebyscheff-Approximation über unbeschränkten Intervallen

A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,

On strong uniqueness in linear complex Chebyshev approximation

wherepn*(f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencenk with the property that the sequence of (nk+2)-point Fekete subsets of

Poles and Alternation Points in Real Rational Chebyshev Approximation

A_{n_k }