Discrepancy of Signed Measures and Polynomial Approximation

Vladimir V. Andrievskii, Hans-Peter Blatt

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English

Springer Verlag
01 June 2002

Mathematics & Sciences; Mathematical foundations; Complex analysis, complex variables

Series: 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.

By: Vladimir V. Andrievskii, Hans-Peter Blatt
Imprint: Springer Verlag
Country of Publication: United States
Volume: Vol 9
Dimensions: Height: 234mm, Width: 156mm, Spine: 25mm
Weight: 1.810kg
ISBN: 9780387986524
ISBN 10: 0387986529
Series: Springer Monographs in Mathematics
Pages: 451
Publication Date: 01 June 2002
Audience: College/higher education , Professional and scholarly , Further / Higher Education , Undergraduate
Format: Hardback
Publisher's Status: Active

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.

Reviews for Discrepancy of Signed Measures and Polynomial Approximation

From the reviews of the first edition: ""Distributions of certain point sets are in many respects important in approximation theory. … Many classical theorems … deal with such problems. This book collects a number of generalizations of these theorems. … The book is written with great care. … This monograph is a valuable addition to the library of any researcher in approximation theory. The topic is rather specialized, but the style and the importance of potential theory in the discipline, makes it also suited for an advanced course in approximation theory."" (Simon Stevin Bulletin, Vol. 11 (1), 2004) ""This book is devoted to discrepancy estimates for the zero of polynomials and for signed measures. … A remarkable feature of the book is that most of the results in it are shown to be sharp. … This work is a valuable monograph on a field that has attracted considerable interest in the recent past, and which has various applications in approximation theory and orthogonal polynomials."" (Vilmos Totik, Bulletin of the London Mathematical Society, Vol. 35, 2003) ""This book discusses in detail the discrepancy of signed measures … . The detailed proofs and the rich reference make this book eligible for a self-study textbook and a reference book, too."" (Béla Nagy, Acta Scientiarum Mathematicarum, Vol. 68, 2002)