The Sinh−1 Transformation in Cardinal Approximation

This paper discusses the Sinc transformation x → φ(x) = sinh−1(x) = log(x + √ 1 + x2), yielding a new class of formulas for approximation and numerical integration on (−∞,∞). Errors decay like O(exp(−cN)) for functions f that are analytic and bounded in the region enclosed by the two branches of the hyperbola {z = x + iy : y/ sin d − x/ cos d = 1}, d < π/2, and for which f has O(|x|−α) decay as |x| → ∞ for interpolation (resp. O(|x|−α−1) decay for quadrature), where α > 0. If f decays like exp(−α|x|), errors decay at the faster rate O(exp(−cN/ logN)). A significant practical value is in the computation of solutions of scattering problems, where integrals containing the Green’s function (4π|~r − ~r ′|)−1 exp(ik|~r − ~r|) must be evaluated. The transformation also yields a new n-point linear approximate of |x| on [−1, 1], 0 < α0 ≤ α ≤ α1, for which the error decreases at a uniform O(exp(−πN logN)) rate.