A new construction is given for approximating a logarithmic potential by a discrete one. This yields a new approach to approximation with weighted polynomials of the form w”n”(” “= uppercase)P”n”(” “= uppercase). The new technique settles several open problems, and it leads to a simple proof for the strong asymptotics on some L p(uppercase) extremal problems on the real line with exponential weights, which, for the case p=2, are equivalent to power- type asymptotics for the leading coefficients of the corresponding orthogonal polynomials. The method is also modified toyield (in a sense) uniformly good approximation on the whole support. This allows one to deduce strong asymptotics in some L p(uppercase) extremal problems with varying weights. Applications are given, relating to fast decreasing polynomials, asymptotic behavior of orthogonal polynomials and multipoint Pade approximation. The approach is potential-theoretic, but the text is self-contained.

Approximation
Logarithmic potentials
Pade approximation
Varying Weight
orthogonal Polynomials

• Front Matter
• Introduction
• Freud weights
• Approximation with general weights
• Varying weights
• Applications
• Back Matter

N/A

Book Title
Weighted Approximation with Varying Weight

Authors
Vilmos Totik

Series Title
Lecture Notes in Mathematics

DOI
https://doi.org/10.1007/BFb0076133

Softcover ISBN
978-3-540-57705-8
Published: 28 February 1994

eBook ISBN
978-3-540-48323-6
Published: 15 November 2006

Series ISSN
0075-8434

Series E-ISSN
1617-9692

Edition Number
1

Number of Pages
VI, 118

Topics
Real Functions, Potential Theory