Abstract. The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi-infinite or an infinite interval, followed by an appropriate approximation procedure on the new region. We first analyse the convergence of these existing methods and show that, in a precisely defined sense, they are sub-optimal. Specifically, they exhibit poor resolution properties, by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as Chebyshev interpolation.To remedy this situation, we introduce two new transforms; one for each of the above settings. We provide full convergence results for these new approximations and then demonstrate that, for a particular choice of parameters, these methods lead to substantially better resolution properties. Finally, we show that optimal resolution power can be achieved by an appropriate choice of parameters, provided one forfeits classical convergence. Instead, the resulting method attains a finite, but user-controlled accuracy specified by the parameter choice.

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Abstract.Chebfun is an established software system for computing with functions of a real variable, but its capabilities for handling functions with singularities are limited. Here an analogous system is described based on sinc function expansions instead of Chebyshev series. This experiment sheds light on the strengths and weaknesses of sinc function techniques. It also serves as a review of some of the main features of sinc methods, including construction, evaluation, zerofinding, optimization, integration, and differentiation.

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This paper discusses modern methods of testing, modeling, and analyzing the dynamics of mechanical structures. The topics covered are mathematical modeling of structural dynamics, modes of vibration, the normal mode versus the transfer function technique for structural testing, and identifying modal parameters from transfer function measurements. As indicated, much of the discussion concerns the modes of vibration of mechanical structures which turn out to be the link between the testing and analysis methods discussed in this paper. These methods are based upon the use of digital computers, and algorithms such as the finite element method for modeling and analysis, and digital filtering such as the fast Fourier transform for dynamic measurements. Some of the latest techniques for graphically displaying modes of vibration using low cost measurement instrumentation will also be presented.