| Discrepancy of Signed Measures and Polynomial Approximation 2002 Edition Contributor(s): Andrievskii, Vladimir V. (Author), Blatt, Hans-Peter (Author) |
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ISBN: 0387986529 ISBN-13: 9780387986524 Publisher: Springer OUR PRICE: $161.49 Product Type: Hardcover - Other Formats Published: December 2001 Annotation: The book is an authoritative and up-to-date introduction to the field of analysis and potential theory dealing with the distribution zeros of classical systems of polynomials such as orthogonal polynomials, Chebyshev, Fekete and Bieberbach polynomials, best or near-best approximating polynomials on compact sets and on the real line. The main feature of the book is the combination of potential theory with conformal invariants, such as module of a family of curves and harmonic measure, to derive discrepancy estimates for signed measures if bounds for their logarithmic potentials or energy integrals are known a priori. |
| Additional Information |
| BISAC Categories: - Mathematics | Probability & Statistics - General - Medical |
| Dewey: 511.4 |
| LCCN: 2001032837 |
| Series: CMS Books in Mathematics |
| Physical Information: 1" H x 6.14" W x 9.21" (1.80 lbs) 438 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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 Szeg 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. |