Extension of a class of periodizing variable transformations for numerical Integration

Class ${\mathcal S}_m$ variable transformations with integer $m$, for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration, Numerical Integration IV, 1993 (H. Brass and G. Hämmerlin, eds.), pp. 359-373.] A representative of this class is the $\sin^m$-transformation that has been used with lattice rules for multidimensional integration. These transformations ``periodize'' the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even $m$. In the present work, we extend these transformations to arbitrary values of $m$, and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitable $m$, they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking $2m$ to be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extended $\sin^m$-transformation.