Asymptotics of Weighted Polynomials
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We survey recent developments in the theory of the weighted polynomials w(x) n P n (x), P n ∈ P n on a closed set A ⊂ R, with a continuous weight w(x) ≥ 0 on A. Important questions are: Where are the extreme points of the weighted polynomials distributed on A, in particular the alternation points of the weighted Chebyshev polynomials w n C w ,n ? Which continuous functions f on A are approximable by the weighted polynomials? How do the polynomials P n of weighted norm w n P n C (A) = 1 behave outside of A?
This is based on our own work (in particular, in the first two sections) and on work of Mhaskar and Saff, and others.
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- Borwein, P. and Saff, E.B., On the denseness of weighted incomplete approximation, these Proceedings.Google Scholar
- M. v. Golitschek, Weierstrass theorem with weights, manuscript, available at the Tampa 1990 Conference.Google Scholar
- G.G. Lorentz, Approximation by incomplete polynomials (problems and results), in: “Padé and Rational Approximations”, E.B. Saff and R.S. Varga, eds., Academic Press, New York, 1977, pp. 289–302.Google Scholar
- G.G. Lorentz, M. v. Golitschek, Y. Makovoz, “Constructive Approximation, Advanced Problems”, book in preparation.Google Scholar
- D.S. Lubinsky and E.B. Saff, Strong asymptotics for extremal polynomials associated with weights on R, Lecture Notes in Math. 1305, Springer, Berlin, 1988.Google Scholar
- E.B. Saff, J.L. Ullman and R.S. Varga, Incomplete polynomials: an electrostatics approach, in: “Approximation Theory, III”, E.W. Cheney, ed., Academic Press, New York, 1980, pp. 769–782.Google Scholar
- M. Tsuji, “Potential Theory in Modern Function Theory”, 2nd edition, Chelsea, New York, 1958.Google Scholar