A generalization of the continuous Euler transformation and its application to numerical quadrature

The continuous Euler transformation, which was proposed by the present author to accelerate the convergence of integrals of Fourier-type, is an analogue of the well-known Euler transformation for series. In this paper, we propose new continuous transformations including a generalization of the continuous Euler transformation. We show that the generalized continuous Euler transformation is applicable to not only a Fourier-type integral but also an oscillatory integral with an algebraically decaying integrand. We also show some numerical examples using the generalized continuous Euler transformation.