Weighted Approximation with Varying Weight - Vilmos Totik

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Résumé :
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 wn( = uppercase)Pn( = 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.

Biographie:
?Edward B. Saff received his B.S. in mathematics from the Georgia Institute of Technology and his Ph.D. from the University of Maryland, where he was a student of the renowned analyst Joseph L. Walsh. Saff's research areas include approximation theory, numerical analysis, and potential theory. He has published more than 290 mathematical research articles, co-authored 9 books, and co-edited 11 volumes. Recognitions of his research include his election as a SIAM Fellow (Society for Industrial and Applied Mathematics) in 2023, as a Foreign Member of the Bulgarian Academy of Sciences in 2013, as a Fellow of the American Mathematical Society in 2013, as well as a Guggenheim Fellowship in 1978. Saff is co-Editor-in-Chief and Managing Editor of the research journal Constructive Approximation and serves on the editorial boards of Computational Methods and Function Theory and the Journal of Approximation Theory. He has mentored 18 Ph.D.'s as well as 13 post-docs. Saff is currently Distinguished Professor of Mathematics at Vanderbilt University. Vilmos Totik was educated in Hungary and was a professor of mathematics at the University of Szeged and the University of South Florida until his retirement. His main research interest is classical mathematical analysis, approximation theory, orthogonal polynomials and potential theory. He has published (partially with co-authors) 5 monographs, one problem book in set theory and about 220 research papers in various disciplines....

Sommaire:
Freud weights.- Approximation with general weights.- Varying weights.- Applications.