3. Electronic Theses and Dissertations (ETDs) - All submissions

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Approximation theory for exponential weights.
(1998) Kubayi, David Giyani.
Much of weighted polynomial approximation originated with the famous Bernstein qualitative approximation problem of 1910/11. The classical Bernstein approximation problem seeks conditions on the weight functions \V such that the set of functions {W(x)Xn};;"=l is fundamental in the class of suitably weighted continuous functions on R, vanishing at infinity. Many people worked on the problem for at least 40 years. Here we present a short survey of techniques and methods used to prove Markov and Bernstein inequalities as they underlie much of weighted polynomial approximation. Thereafter, we survey classical techniques used to prove Jackson theorems in the unweighted setting. But first we start, by reviewing some elementary facts about orthogonal polynomials and the corresponding weight function on the real line. Finally we look at one of the processes (If approximation, the Lagrange interpolation and present the most recent results concerning mean convergence of Lagrange interpolation for Freud and Erdos weights.