Almost-interpolatory Chebyshev quadrature

Abstract:

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an n-point formula be at least equal to n. This condition may be expressed as ${\left \| d \right \|_\nu } = 0,1 \leqq p$, where $d = ({d_1}, \cdots ,{d_n})$ with \[ {d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,} \] ${\mu _j}(\omega ),j = 0,1, \cdots$, are the moments of the weight function $\omega$ used in the quadrature, and ${x_1}, \cdots ,{x_n}$ are the nodes. In those cases when ${\left \| d \right \|_2}$ does not vanish for a real choice of nodes, it has been proposed that a real minimizer of ${\left \| d \right \|_2}$ be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of ${\left \| d \right \|_p},1 \leqq p < \infty$, always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.