Univariate Interpolation on a Regular Finite Grid by a Multiquadric Plus a Linear Polynomial

Univariate multiquadric interpolation to a twice continuouslydifferentiable function on a regular infinite grid enjoys secondorder convergence and some excellent localization properties,but numerical calculations suggest that, if the grid is finite,then usually the convergence rate deteriorates to first ordernear the grid boundaries, ibis conjecture is proved. It is alsoshown that one can recover superlinear convergence by addinga linear polynomial term to the multiquadric approximation.Making such additions is a standard technique, but we find thatthe usual way of choosing the polynomial fails to provide superlinearconvergence m general. Therefore some new procedures are giventhat pick a suitable polynomial automatically. Thus it is notunusual to reduce the maximum error of the interpolation bya factor of 10³. Further, it is straightforward to include oneof the new procedures in multiquadric interpolation to functionsof several variables when the data points are in general position.