Discrepancy of Signed Measures and Polynomial Approximation

1. Auflage. Softcover version of original hardcover Auflage 2002
ISBN: 978-1-4419-3146-7
Verlag: Springer

Buch, Englisch, 438 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 1400 g

Reihe: Springer Monographs in Mathematics

In many situations in approximation theory the distribution of points in a given set is of interest. For example, the suitable choiee of interpolation points is essential to obtain satisfactory estimates for the convergence of interpolating polynomials. Zeros of orthogonal polynomials are the nodes for Gauss quadrat ure formulas. Alternation points of the error curve char­ acterize the best approximating polynomials. In classieal complex analysis an interesting feature is the location of zeros of approximants to an analytie function. In 1918 R. Jentzsch [91] showed that every point of the circle of convergence of apower series is a limit point of zeros of its partial sums. This theorem of Jentzsch was sharpened by Szegö [170] in 1923. He proved that for apower series with finite radius of convergence there is an infinite sequence of partial sums, the zeros of whieh are "equidistributed" with respect to the angular measure. In 1929 Bernstein [27] stated the following theorem. Let f be a positive continuous function on [-1, 1]; if almost all zeros of the polynomials of best 2 approximation to f (in a weighted L -norm) are outside of an open ellipse c with foci at -1 and 1, then f has a continuous extension that is analytic in c.

Zielgruppe

Research

Autoren/Hrsg.

Blatt, Hans-Peter

Andrievskii, Vladimir V.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik

Inhaltsverzeichnis

1 Auxiliary Facts.- 2 Zero Distribution of Polynomials.- 3 Discrepancy Theorems via Two—Sided Bounds for Potentials.- 4 Discrepancy Theorems via One-Sided Bounds for Potentials.- 5 Discrepancy Theorems via Energy Integrals.- 6 Applications of Jentzsch—Szegö and Erdös—Turán Type Theorems.- 7 Applications of Discrepancy Theorems.- 8 Special Topics.- A Conformally Invariant Characteristics of Curve Families.- A.1 Module and Extremal Length of a Curve Family.- A.2 Reduced Module.- B Basics in the Theory of Quasiconformal Mappings.- B.1 Quasiconformal Mappings.- B.2 Quasiconformal Curves and Arcs.- C Constructive Theory of Functions of a Complex Variable.- C.1 Jackson Type Kernels.- C.2 Polynomial Kernels Approximating the Cauchy Kernel.- C.3 Inverse Theorems.- C.4 Polynomial Approximation in Domains with Smooth Boundary.- D Miscellaneous Topics.- D.1 The Regularized Distance.- D.2 Green’s Function for a System of Intervals.- Notation.

Über Autor(innen)

Vladimir V. Andrievskii is Assistant Professor of Mathematics at Kent State University. Hans-Peter Blatt is Full Professor of Mathematics at Katholische Universität Eichstätt.