Approximation theory for exponential weights.
Date
1998
Authors
Kubayi, David Giyani.
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Abstract
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.
Description
Keywords
Approximation theory., Orthogonal polynomials., Polynomials.