New error coefficients for estimating quadrature errors for analytic functions

Abstract:

Since properly normalized Chebyshev polynomials of the first kind ${\tilde T_n}(Z)$ satisfy \[ ({\tilde T_m},{\tilde T_n}) = \int _{ \in \rho } {{{\tilde T}_m}({\text {z}})} \overline {{T_n}({\text {z}})} |1 - {{\text {z}}^2}{|^{ - 1/2}}|d{\text {z}}| = {\delta _{mn}}\] for ellipses $\in \rho$ with foci at $\pm 1$, any function analytic in $\in \rho$ has an expansion, $f({\text {z}}) = \sum {{a_n}{{\tilde T}_n}({\text {z}})}$ with ${a_n} = (f,{\tilde T_n})$. Applying the integration error operator $E$ yields $E(f) = \sum {{a_n}E({{\tilde T}_n})}$. Applying the Cauchy-Schwarz inequality to $E(f)$ leads to the inequality \[ |E(f){|^2} \leqq \sum {|{a_n}{|^2}} \sum {|E({{\tilde T}_n})} {|^2} = ||f|{|^2}||E|{|^2}.\] . $||E||$ can be computed for any integration rule and approximated quite accurately for Gaussian integration rules. The bound for $|E(f)|$ using this norm is compared to that using a previously studied norm based on Chebyshev polynomials of the second kind and is shown to be superior in practical situations. Other results involving the latter norm are carried over to the new norm.