Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: The \(L_ \infty\) theory (Q1198158)

This paper gives sufficient conditions for the weighted \(L^ \infty\) convergence of Hermite-Féjer and Hermite interpolation polynomials using weights of the form \(W^ 2=e^{-2Q}\), where \(Q\) is a ``nice'' function of at least polynomial growth, in the definition of the Hermite- Féjer and Hermite polynomials, and weights of the form \(W^ 2(1+| Q'|)^{-K}(1+| x|)^{-1}\) for the \(L^ \infty\) convergence. The result is general enough to include many natural weight functions. The paper contains many technical estimates with natural interpretations of these estimates, and some of these micht be useful in further investigations. There is a section of quadrature sum estimates that may be of interest apart from the main thrust of the paper.