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SAT 3 (2007), 1
Surveys in Approximation Theory, 3 (2007), 1-105.
A Survey of Weighted Polynomial Approximation
with Exponential Weights
Doron Lubinsky
Abstract. Let W: R → (0,1] be
continuous. Bernstein's approximation problem, posed in 1924, deals with
approximation by polynomials in the weighted uniform norm f →
||fW|| L∞(R) . The qualitative form
of this problem
was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative
forms of the problem were actively investigated starting from the 1960's. We
survey old and recent aspects of this topic, including the Bernstein
problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and
Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We
present the main ideas used in many of the proofs, and different techniques
of proof, though not the full proofs. The class of weights we consider is
typically even, and supported on the whole real line, so we exclude Laguerre
type weights on [0,∞). Nor do we discuss Saff's weighted
approximation problem, nor the asymptotics of orthogonal polynomials.
E-print: arXiv:0701099
Published: 1 January 2007.
Doron Lubinsky
School of Mathematics
Georgia Institute of Technology
Atlanta, GA, 30332-0160
E-mail: lubinsky@math.gatech.edu