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Weighted Approximation with Varying Weight
English
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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.
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