Periodizing transformations for numerical integration

The order of the trapezoidal rule can be raised by making a substitution that transforms the integrand into a new integrand which has a smooth periodic extension. The exponent in the rate at which the transformed weights approach zero near the endpoints is called the damping power of the transformation. Currently used polynomial transformations of damping power m have order m+1 if m is odd, and order m+2 if m is even. We prove that the highest possible order is m+1 if m is odd, but 2m+2 if m is even, and give new polynomial transformations that achieve the optimal order.