ACTA issues

Converse quadrature sum inequalities for polynomials with Freud weights

D. S. Lubinsky

Acta Sci. Math. (Szeged) 60:3-4(1995), 527-557

Abstract. We investigate converse quadrature sum (or Marcinkiewicz--Zygmund) inequalities of the form $$\int_{-\infty}^\infty |PW|^p(x)\bigl(1+|x|\bigr)^{rp}dx\le C\sum_{j=1}^n \lambda_{jn}|PW|(x_{jn})^p W^{-2}(x_{jn})\bigl(1+|x_{jn}|\bigr)^{Rp} $$ for all polynomials $P$ of degree $\le n-1$ and $n\ge1$, with $C$ independent of $n$ and $P$. Here the $\{x_{jn}\}$ and $\{\lambda_{jn}\}$ are the Gauss points and weights for the Freud weight $W^2$ (for example, $W_\beta(x) :=\exp\left(-\frac12 |x|^\beta\right)$, $\beta >1$) and $r,R\in{\msbm R}$, $1< p< \infty$. We derive necessary and almost matching sufficient conditions for such inequalities that may be applied at least for $W^2=W_\beta^2$, $\beta >1$.

AMS Subject Classification (1991): 41A55, 65D04, 42C05

Keyword(s): Gauss quadrature, quadrature sums, polynomials, Freud weights, Lagrange interpolation, orthogonal expansions

Received July 27, 1994. (Registered under 5649/2009.)