Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms

We present five theorems concerning the asymptotic convergence rates of Chebyshev interpolation applied to functions transplanted to either a semi-infinite or an infinite interval under exponential or double-exponential transformations. This strategy is useful for approximating and computing with functions that are analytic apart from endpoint singularities. The use of Chebyshev polynomials instead of the more commonly used cardinal sinc or Fourier interpolants is important because it enables one to apply maps to semi-infinite intervals for functions which have only a single endpoint singularity. In such cases, this leads to significantly improved convergence rates

Error analyses of Sinc-collocation methods for exponential decay initial value problems

Nurmuhammad et al. developed Sinc-Nystr\"{o}m methods for initial value problems in which solutions exhibit exponential decay end behavior. In the methods, the Single-Exponential (SE) transformation or the Double-Exponential (DE) transformation is combined with the Sinc approximation. Hara and Okayama improved those transformations so that a better convergence rate could be attained, which was afterward supported by theoretical error analyses. However, due to a special function included in the basis functions, the methods have a drawback for computation. To address this issue, Okayama and Hara proposed Sinc-collocation methods, which do not include any special function in the basis functions. This study gives error analyses for the methods.Comment: Keywork: Ordinary differential equations, Initial value problems, Volterra integral equations, Sinc numerical methods, SE transformation, DE transformatio

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators