Discrepancy of Signed Measures and Polynomial Approximation (Springer Monographs in Mathematics)

Andrievskii, Vladimir V. and Hans-Peter Blatt:

ISBN 10: 0387986529 / ISBN 13: 9780387986524

Editore: Springer, 2002

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438 Seiten ex Library Book aus einer wissenschafltichen Bibliothek Altersfreigabe FSK ab 0 Jahre Sprache: Englisch Gewicht in Gramm: 969 23,6 x 15,2 x 2,5 cm, Gebundene Ausgabe.

Codice articolo 336017

Dati bibliografici

• Titolo: Discrepancy of Signed Measures and Polynomial Approximation (Springer Monographs in Mathematics)
• Editore: Springer
• Data di pubblicazione: 2002
• Lingua: Inglese
• ISBN 10: 0387986529
• ISBN 13: 9780387986524
• Rilegatura: Rilegato
• Edizione: Auflage: 2002.
• Condizione: Gut
• Cataloghi dei venditori:
  • Alle Bücher

Descrizione articolo

Riassunto

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.

Dalla quarta di copertina

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. Classical results of Jentzsch and Szegö for the zero distribution of partial sums of power series can be recovered and sharpened by new discrepany estimates, as well as distribution results of Erdös and Turn for zeros of polynomials bounded on compact sets in the complex plane.
Vladimir V. Andrievskii is Assistant Professor of Mathematics at Kent State University. Hans-Peter Blatt is Full Professor of Mathematics at Katholische Universität Eichstätt.