New error coefficients for estimating quadrature errors for analytic functions

Authors:
Philip Rabinowitz and Nira Richter

Journal:
Math. Comp. **24** (1970), 561-570

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275675-X

MathSciNet review:
0275675

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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.

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Keywords:
Error coefficients,
error in numerical integration,
analytic functions,
Chebyshev polynomials,
complete orthonormal set,
error estimates,
trapezoidal rule,
Simpson rule,
norm of error functional,
interpolatory quadrature

Article copyright:
© Copyright 1970
American Mathematical Society