Rational approximation, oscillatory Cauchy integrals and Fourier transforms

摘要

We develop the convergence theory for a well-known method for theinterpolation of functions on the real axis with rational functions. Precisenew error estimates for the interpolant are de- rived using existing theory fortrigonometric interpolants. Estimates on the Dirichlet kernel are used toderive new bounds on the associated interpolation projection operator. Errorestimates are desired partially due to a recent formula of the author for theCauchy integral of a specific class of so-called oscillatory rationalfunctions. Thus, error bounds for the approximation of the Fourier transformand Cauchy integral of oscillatory smooth functions are determined. Finally,the behavior of the differentiation operator is discussed. The analysis herecan be seen as an extension of that of Weber (1980) and Weideman (1995) in amodified basis used by Olver (2009) that behaves well with respect to functionmultiplication and differentiation.