Norm Bounds of Quadrature Processes

Abstract

A quadrature process whose norm does not remain bounded with increasing order, is divergent. In this note it is shown that for an nth order formula the norm may be estimated by $\| {R_n } \| \geqq 2^n | {R_n (x^{n + 1} )} |$. From this result we infer that the quadrature process using the zeros of the Jacobi polynomial $P_{n + 1}^{(\alpha ,\beta )} $ as nodes (with $\max (\alpha ,\beta ) > \frac{2}{3}$) and the Newton–Cotes process diverge. We emphasize that our method is very elementary in manner and does not contain the technical difficulties found in the usual proofs.