Discrepancy of Signed Measures and Polynomial Approximation

1. Front Matter

   Pages i-xiii

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2. Auxiliary Facts

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 1-48

3. Zero Distribution of Polynomials

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 49-93

4. Discrepancy Theorems via Two-Sided Bounds for Potentials

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 95-128

5. Discrepancy Theorems via One-Sided Bounds for Potentials

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 129-172

6. Discrepancy Theorems via Energy Integrals

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 173-186

7. Applications of Jentzsch-Szegő and Erdős-Turán Type Theorems

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 187-224

8. Applications of Discrepancy Theorems

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 225-319

9. Special Topics

   • Vladimir V. Andrievskii, Hans-Peter Blatt

   Pages 321-340

10. Back Matter

    Pages 341-438

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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.

• Approximation Theory
• Complex Analysis
• Constructive Analysis
• Geometric Function Theory
• Invariant
• Polynomials
• Potential Theory
• Variable
• calculus
• function
• theorem

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)

• Department of Mathematics and Computer Science, Kent State University, Kent, USA

  Vladimir V. Andrievskii

• Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Eichstätt, Germany

  Hans-Peter Blatt

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