Weighted Approximation with Varying Weight

1. Front Matter

   Pages I-VI

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2. Introduction

   • Vilmos Totik

   Pages 1-5

3. Freud weights

   • Vilmos Totik

   Pages 7-20

4. Approximation with general weights

   • Vilmos Totik

   Pages 21-48

5. Varying weights

   • Vilmos Totik

   Pages 49-77

6. Applications

   • Vilmos Totik

   Pages 79-110

7. Back Matter

   Pages 111-115

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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

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