Overview
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.
Full Product Details
Author: Vladimir V. Andrievskii ,
Hans-Peter Blatt
Publisher: Springer-Verlag New York Inc.
Imprint: Springer-Verlag New York Inc.
Edition: Softcover reprint of hardcover 1st ed. 2002
Dimensions:
Width: 15.50cm
, Height: 2.30cm
, Length: 23.50cm
Weight: 1.400kg
ISBN: 9781441931467
ISBN 10: 1441931465
Pages: 438
Publication Date: 06 December 2010
Audience:
Professional and scholarly
,
Professional and scholarly
,
Professional & Vocational
,
Professional & Vocational
Format: Paperback
Publisher's Status: Active
Availability: In Print
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Reviews
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. (Bela Nagy, Acta Scientiarum Mathematicarum, Vol. 68, 2002)
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)
"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)"
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. (Bela Nagy, Acta Scientiarum Mathematicarum, Vol. 68, 2002)
