Almost - Interpolatory Chebyshev Quadrature

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 npoint formula be at least equal to n. This condition may be expressed as | \d\ \p = 0, 1 g p, where d (dx, ■ ■ ■ , d„) with Mo(w) ~ , -IT dj = 2w A iM ; = 1, 2, • • ■ , z!, ZJ ,_, Pj(io), j = 0, 1, • • • , are the moments of the weight function u used in the quadrature, and xi, ■ ■ ■ , x„ are the nodes. In those cases when | \d\ \i does not vanish for a real choice of nodes, it has been proposed that a real minimizer of | \d\ |2 be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of ||rf||,,, 1 â P < <=, 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.