Eli Levin's research while affiliated with The Open University of Israel and other places

The smoothness of σt plays a role in discretising the potential to obtain weighted polynomial approximations. In this chapter, we establish various levels of smoothness of σt under corresponding conditions on Q.

In this chapter, we obtain further upper (and lower) bounds on orthogonal polynomials and on their L p norms. We also estimate fundamental polynomials of Lagrange interpolation, and spacing of zeros of orthogonal polynomials. We shall often need more than W∈F(lip1/2). Recall from Chapter 1 that we defined W∈F(lip1/2+) if both W∈F(lip1/2) and for each L>1, there exists C>0 and t 0 such that $$ Q'\left( {{a_{{Lt}}}} \right)/Q'\left( {{a_{t}}} \right) \geqslant 1 + C,\left| t \right| \geqslant {t_{0}}. "

We have already seen that for Q convex, and not identically vanishing polynomials P of degree ≦ n, there holds the Mhaskar-Saff inequality: $$ \left| {P{e^{{ - Q}}}} \right|(x) < {\left\| {PW} \right\|_{{{L_{\infty }}({\Delta _{n}})}}},x \in I\backslash {\Delta _{n}}. "

In this chapter, we establish pointwise asymptotics of the orthonormal polynomials p n (W 2, x) for x in the interval of orthogonality, as well as asymptotics for the associated recurrence coefficients. We shall also reformulate some of the results of the previous chapters for this special case.

Perhaps the most significant result of this work is the following uniform bound on the orthogonal polynomials throughout the interval of orthogonality:

Recall that given a weight W2 on the finite or infinite interval I = (c, d), its nth orthonormal polynomial pn (W2,x) has zeros \( \left\{ {{x_{{jn}}}} \right\}\begin{array}{*{20}{c}} n \hfill \\ {j = 1} \hfill \\ \end{array} , \), where $$ c < {x_{{nn}}} < {x_{{n - 1,n}}} < \cdot \cdot \cdot < {x_{{2n}}}{ < _{{1n}}} < d. "

In this chapter, we obtain upper and lower bounds for the equilibrium density σt (x) and for the associated potential V μt . The lower bounds are easy to obtain, and minimal assumptions on W are needed:

In this chapter, we establish asymptotics of the extremal errors $$ {E_{{n,p}}}(W): = \mathop{{\inf }}\limits_{{P(x) = {x^{n}} + \cdot \cdot \cdot }} \left\| {PW} \right\|{L_{p}}(I), "$$ where the inf is taken over all monic polynomials P of degree n. In order to state our result in compact form, we need some notation. For a non-negative h : [−1,1] → ℝ, let $$ G\left[ h \right]: = \exp \left( {\frac{1}{\pi }\int_{{ - 1}}^{1} {\frac{{\log h(u)}}{{\sqrt {{1 - {u^{2}}}} }}du} } \right) "$$ denote the geometric mean of h. Recall also that $$ W_{n}^{{_{*}}}\left( u \right): = W\left( {L_{n}^{{\left[ { - 1} \right]}}\left( u \right)} \right),{\text{ u}} \in {\text{Ln}}\left( {\text{I}} \right){\text{,}} "$$ where L n is the linear map of [a −n, an] onto [−1,1] and L n[−1] is its inverse. Finally, let $$ {\kappa _{p}}: = \left\{ {\begin{array}{*{20}{c}} {{{\left( {\sqrt {\pi } \Gamma \left( {\frac{{p + 1}}{2}} \right)/\Gamma \left( {\frac{p}{2} + 1} \right)} \right)}^{{1/p}}}} \hfill \\ {1,} \hfill \\ \end{array} } \right.\begin{array}{*{20}{c}} {p < \infty } \\ {p = \infty } \\ \end{array} . "

In this chapter, we establish some basic estimates for Q, a t , and similar quantities. We shall deal with various classes of weights on I=(c, d). Recall that I may be finite or infinite and contains 0 as an interior point. We shall often state results only for the interval (0, d); the reader will easily observe what is the analogous statement for (c, 0).

We present some of the basics of weighted potential theory, and derive useful identities involving a t .