Error Bounds for Asymptotic Expansions of Integrals
Received by the editors March 30, 1979, and in revised form October 24, 1979.
The purpose of this paper is to give an up-to-date account of the error analysis of asymptotic expansions of integral transforms. Particular attention is paid to two new asymptotic tools that have been developed recently, one based on summability methods, the other based on the use of distributions. After a brief discussion of integrals with exponentially decaying kernels, we first turn to the consideration of integral transforms whose kernels are oscillatory. This includes the Fourier, Hankel and multi-dimensional Fourier transforms. Next, we consider the class of integral transforms whose kernels are algebraic functions, e.g., Stieltjes, Hilbert and fractional integral transforms. Also, a quite thorough treatment is given to a Mellin-type convolution integral in which both the function and the kernel are algebraically dominated at t = 0+ and t =+∞. Finally, we construct error bounds for the remainders associated with two uniform asymptotic expansions, one for the coalescence of a stationary point and an end point, the other for the coalescence of two stationary points. These bounds are unfortunately difficult to evaluate, and attention is being called to develop completely new methods. Another problem is proposed at the end of the paper, which concerns error bounds for certain asymptotic expansions involving logarithms.
