| Discrepancy of Signed Measures and Polynomial Approximation Contributor(s): Andrievskii, Vladimir V. (Author), Blatt, Hans-Peter (Author) |
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ISBN: 1441931465 ISBN-13: 9781441931467 Publisher: Springer OUR PRICE: $161.49 Product Type: Paperback - Other Formats Published: December 2010 |
| Additional Information |
| BISAC Categories: - Medical - Mathematics | Probability & Statistics - General - Mathematics | Mathematical Analysis |
| Dewey: 511.4 |
| Series: Springer Monographs in Mathematics |
| Physical Information: 0.92" H x 6.14" W x 9.21" (1.40 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. |