Asymptotics of Extremal Errors

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, a_n] onto [−1,1] and L_n[−1] is its inverse. Finally, let $$ {\kappa _{p}}: = \left\{ {\begin{array}{*{20}{c}} {{{\left( {\sqrt {\pi } \Gamma \left( {\frac{{p + 1}}{2}} \right)/\Gamma \left( {\frac{p}{2} + 1} \right)} \right)}^{{1/p}}}} \hfill \\ {1,} \hfill \\ \end{array} } \right.\begin{array}{*{20}{c}} {p < \infty } \\ {p = \infty } \\ \end{array} . "$$